{"text":"\\section{Introduction.}\nIn this article we introduce a family of quantities, denoted by ${\\mathcal{M}}_{a}$ (where $a$, an arbitrary real number, is the parameter of the family) naturally attached to (integrable) geodesic congruences ${\\mathcal{F}}$, of Static Solutions of the Einstein Equations in dimension three. The invariants (which can be seen as a real functions over the range of the congruence) are shown to be monotonic along each of the geodesics of ${\\mathcal{F}}$. Moreover whenever ${\\mathcal{M}}_{a}$ is stationary along a geodesic $\\gamma$ of ${\\mathcal{F}}$, then the local geometry along $\\gamma$ can be seen to be of Schwarzschild form. In this sense ${\\mathcal{M}}_{a}$ measures a certain departure of the given static solution to the Schwarzschild solution. The framework that we will develop out of these invariants\nis a natural extension of the standard comparison techniques of Riemannian spaces of non-negative Ricci curvature. However, as we incorporate into ${\\mathcal{M}}_{a}$ the influence that the lapse exerts on the Ricci curvature, and, as a result, the monotonicity of ${\\mathcal{M}}_{a}$ sharply captures the departure from the Schwarzschild solution (not from the Euclidean space), the framework here developed can be best described as one that compares static solutions to the Schwarzschild solution.  It is thus not peculiar that when the technique is applied to asymptotically flat static solutions with regular and connected horizons, the uniqueness of the Schwazschild solution is achieved with remarkable naturalness. It is worth noting that the novel proof of this central result in General Relativity that we shall provide does not require the intermediate step of proving conformal flatness of previous proofs.  The ideas that we will describe can be interpreted as partial results on the bigger proposal of developing a more complex comparison theory for static solutions in arbitrary dimensions. \n\nBefore continuing with the description of the contents, we briefly introduce static solutions of the Einstein equations and summarize some properties that would place the contents into an adequate perspective.\n\n\\subsection{Elements of static solutions.}\n\nA static solution of the Einstein equations in dimension three \\footnote{In this article we will restrict to dimension three. Our most important invariant, the quantity ${\\mathcal{M}}$ (see later), is monotonic only in dimension three and we do not know, at the moment, a replacement of it to higher dimensions. The static Einstein equations (\\ref{SE1})-(\\ref{SE2}) are valid in any dimension.}, is given by a triple $(\\Sigma,g,N)$ where $\\Sigma$ is a smooth Riemannian three manifold possibly with boundary, $g$ is a smooth Riemannian metric and $N$, the {\\it Lapse Function}, is a smooth function, strictly positive in ${\\rm int (}\\Sigma{\\rm )}$, and satisfying\n\\begin{align}\\label{SE1}\nN Ric&=\\nabla\\nabla N,\\\\\n\\label{SE2}\n\\Delta N&=0.\n\\end{align}\n\n\\noindent These equations, note, are invariant under simultaneous but independent scalings on $g$ and $N$. \n\nThe description of static solutions is better separated into local and global properties. From the local point of view, the geometry of static solutions is controlled in $C^{\\infty}$ by two weak invariants. This is a direct consequence of Anderson's curvature estimates ~\\cite{MR1809792} (applying in dimension three) which are described as follows. Let $(\\Omega,g,N)$ be a static solution of the Einstein equations, where $(\\Omega,g)$ is a complete Riemannian manifold with or without boundary. Then there is a universal constant $K>0$ such that for any $p\\in\\Omega$ we have\n\\begin{equation}\n|Rm|+|\\nabla \\ln N|^{2}\\leq   \\frac{K}{dist(p,\\partial \\Omega)^{2}},\n\\end{equation}\n\n\\noindent where if $\\partial \\Omega=\\emptyset$ we set $dist(p,\\partial \\Omega)=\\infty$. Note that this shows in particular that the only complete and boundary-less static solution in dimension three is covered (after normalizing $N$ to one) by the trivial solution $(\\field{R}^{3},g_{\\field{R}^{3}},N=1)$. Anderson's curvature estimates together with the Bishop-Gromov volume comparison and standard elliptic estimates, imply the following {\\it interior estimates} for static solutions in dimension three.\n\n\\begin{Lemma} {\\rm {\\bf (Interior's estimates (Anderson)})} Let $\\Omega$ be a closed three-dimensional manifold with non-empty boundary $\\partial \\Omega$. Suppose that $(\\Omega,g,N)$ is a static solution of the Einstein equations. Let $p\\in \\Omega$, let $d=dist(\\partial \\Omega)$ and let $V_{1}=Vol(B(p,d_{1}))$ for $d_{1}<d$. Then there is $d_{2}(d,d_{1},V_{1})>0$, and for any $i\\geq 0$ there are $\\Lambda(d,d_{1},V_{1},i)>0$, $I(i,d_{1},V_{1})>0$,  such that $inj(p)\\geq I$ and $\\| \\nabla ^{i}Rm\\|_{L^{\\infty}_{g}(B(p,d_{2}))}\\leq \\Lambda$.\n\\end{Lemma}\n\n\\noindent These interior estimates, in turn imply, as is well known, the control of the $C^{i}_{\\{x_{j}\\}}$ norm of the entrances $g_{ij}$ of $g$, in suitable harmonic coordinates $\\{x_{j}\\}$ covering $B(p,d_{2})$, and from them precompactness statements can be obtained.\n\nThe global geometry of static solutions instead is greatly influenced by boundary conditions and, in many cases, boundary conditions provide uniqueness. This occurs when, for instance, one assumes that $\\partial \\Sigma$ consist of a finite set of {\\it regular horizons} plus further hypothesis on the asymptotic of $(\\Sigma,g)$ at infinity. We will adopt the following definition (see \\cite{MR1809792}).\n\n\\begin{Definition}\nThe boundary $\\partial \\Sigma$ of the smooth manifold $\\Sigma$ is a {\\it regular horizon} iff $\\partial \\Sigma$ is a finite union of compact (boundary-less) surfaces $H_{i},i=1,\\ldots n$, $\\partial \\Sigma=\\{q\/N(q)=0\\}$ and at each $H_{i}$ we have $|\\nabla N|\\bigg|_{H_{i}}>0$.\n\\end{Definition}\n\n\\noindent It follows easily from the static equations (\\ref{SE1})-(\\ref{SE2}) that every regular horizon $\\partial \\Sigma$ is totally geodesic and $|\\nabla N|$ is constant and different from zero on each component. \n\nPerhaps the easiest examples of complete solutions with regular horizons are the {\\it Flat solutions} that we will denote by the triple $(\\Sigma_{F},g_{F},N_{F})$. They have the presentation \n\\begin{equation}\n\\Sigma_{F}=[0,\\infty)\\times T^{2},\\ N_{F}=r,\\ g_{F}=dr^{2}+h_{F},\n\\end{equation}\n\n\\noindent where $h_{F}$ is a flat metric in $T^{2}$. The family is parameterized by the set of flat metrics in $T^{2}$ (non-isometric). Note that we have demanded that $N$ grows linearly with respect to arc length and with slope one. Of course any $N$ that grows linearly can be scaled to have growth of slope one.  \n  \nYet, the prototypical and central examples of static metrics are the Schwarzschild solutions. Recall, the Schwarzschild solution $(\\Sigma_{N},g_{S},N_{S})$ of mass $m\\geq 0$ has the presentation\n\\begin{equation}\\label{SchP} \n\\Sigma_{S}=[2m,\\infty)\\times S^{2},\\ N_{S}=\\sqrt{1-\\frac{2m}{r}},\\ g_{S}=dr^{2}+r^{2}(1-\\frac{2m}{r})d\\Omega^{2},\n\\end{equation}\n\n\\noindent while if $m<0$ the presentation\n\\begin{equation}\\label{SchP2} \n\\Sigma_{S}=(0,\\infty)\\times S^{2},\\ N_{S}=\\sqrt{1-\\frac{2m}{r}},\\ g_{S}=dr^{2}+r^{2}(1-\\frac{2m}{r})d\\Omega^{2}.\n\\end{equation}\n\n\\noindent The ``uniqueness of the Schwazschild solution\", as in known today and in the form presented below, came as the result of several efforts, starting from the seminal work of Israel in 1967. For the history of the developments which lead to the proof of this important result as well as accurate references we refer to the article \\cite{Robinson}. \n\n\\begin{Theorem} {\\rm (Schwarzschild's uniqueness ~\\cite{Israel}, ~\\cite{RobinsonII},~\\cite{MR876598})}\\label{TRS}\nLet $(\\Sigma,g,N)$ be a static solution of the Einstein equations of dimension three. Suppose it is asymptotically flat (with one end) and with regular, possibly empty, and possibly disconnected horizon $\\partial \\Sigma$. Then the solutions is a Schwarzschild solution of non-negative mass. \n\\end{Theorem}\n\nSeveral hypothesis of this theorem can be relaxed still obtaining the same uniqueness outcome. \nFor instance suppose there is one end but the hypothesis of asymptotic flatness, or even the topological nature of the end, is withdrawn, then results exist showing that the solution is still one of the Schwazschild family of positive mass. In particular when $N\\leq N_{0}<\\infty$ but nothing of the end is known, not even the a priori topology, then it can be shown\n\\footnote{This follows from a combination of results. First observe that $N$ cannot go uniformly to zero over the end, for in such case, as $N$ is harmonic and is zero over the horizon, we would violate the maximum principle. Using the notation in ~\\cite{MR1809792} denote by $t(p)$ the $g$-distance from a point $p$ to the horizon $H$. Denote also by $B(H,\\bar{t})$ the ball of center $H$ and radius $\\bar{t}$, namely $B(H,\\bar{t})=\\{p\/ t(p)<\\bar{t}\\}$. Now, from Theorem 0.3 (ii) in\\cite{MR1809792}, either the end is asymptotically flat or small in the sense that $\\int^{\\infty}\\frac{1}{A(\\partial B(H,\\bar{t}))}d\\bar{t}=\\infty$. Assume $N\\leq N_{0}$. Consider $f=N_{0}+1-N$. Then $\\Delta f=0$ and $\\Delta \\ln f=-|\\nabla \\ln f|^{2}$. Define $F(t):=\\int_{B(H,t)\\setminus B(H,\\bar{t}_{1})}|\\nabla \\ln f|^{2}dV$. For $t_{1}$ small, we have $\\int_{\\partial B(H,t_{1})}g(\\nabla \\ln f,n_{in})dA>0$, where $n_{in}$ is the unit normal to $\\partial B(H,t_{1})$ pointing inwards to the ball. Using this fact, integrating $\\Delta \\ln f=-|\\nabla \\ln f|^{2}$ over $B(H,t)\\setminus B(H,\\bar{t}_{1})$ and using Cauchy-Schwarz one easily deduce the inequality $F'\/F^{2}\\geq 1\/A$. From it one gets \n$1\/F(t)\\leq 1\/F(t_{2})-\\int_{t_{2}}^{t}\\frac{1}{A}d\\bar{t}$, where $t_{2}>t_{1}$. Thus if the end is small, one would get $F=\\infty$ at a finite distance form $H$, which is not possible.} that the solution is indeed a Schwarzschild solution. The same occurs when it is known that outside a compact set, each end is homeomorphic to $\\field{R}^{3}$ minus a ball\\footnote{arXiv:1002.1172} and over there the metric $N^{2}g$ is complete, which occurs for example when $N\\geq N_{0}>0$. In all these generalizations, which are important for deeper understanding of Einstein's theory, it is assumed that the space $(\\Sigma,g)$, as a metric space, is complete. \n\nWe feel that the following broader conjecture may be accessible.\n\\begin{Conjecture}  Let $(\\Sigma,g,N)$ be a complete solution of the Static Einstein equations with regular but possibly disconnected (non-empty) horizon $\\partial \\Sigma$. Suppose that the conformal metrics $N^{2}g$ and $N^{-2}g$ are complete outside given domains of compact closure on each end of $(\\Sigma,g)$. Then the solution is either a Schwarschild solution or a flat solution.\n\\end{Conjecture} \n\n\\noindent Observe that no assumption is made on the topology of the ends.\n   \nWhen boundary data is prescribed, and is not the data of a regular horizon, and the hypothesis of asymptotic flatness is kept, then much less is known about the existence of solutions although a conjecture \\cite{MR1957036} and partial results do exist\\footnote{arXiv:0909.4550} under some hypothesis.  In whatever case, Dirichlet-type of problems for the Einstein equations are interesting from physical and mathematical reasons. A theory, a highly necessary task, is still lacking.  \n\nThe Schwarzschild family is unique, but, why?. Are the present proofs satisfactory as an answer to this question?. Do we need to place the problem of the uniqueness of the Schwarzschild family into a larger one to understand it better?. Which one would be that bigger perspective?. Could it be a Dirichlet-type of theory for the Static Einstein equations?. Despite all the accumulated knowledge, some aspects of the uniqueness of the Schwarzschild solutions remains (to us) somehow mysterious. The present work would try to clarify the phenomenon from the perspective of comparison geometry. It is worth finally to remark that there are yet further reasons of why it is important to have different proofs and points of view regarding Theorem \\ref{TRS}. Just mention that the elusive and yet inconclusive notions of localized energy or the even more conjectural notion of entropy may have to do and could be better clarified with different understandings of the Schwazschild uniqueness.\n    \n\\subsection{${\\mathcal{M}}_{a}$ and comparison geometry.}    \n\n\\vs\nThe idea underlying the technique that we will describe is rather simple. First, and most important, we will work in the {\\it harmonic map representation} of static solutions. Namely, instead of working with the variables $(g,N)$ we will work with the variables $({\\tt g},{\\tt N})=(N^{2}g,\\ln N)$. The Einstein equations (\\ref{SE1})-(\\ref{SE2}) now become \n\\begin{align}\n{\\tt Ric}&=2d {\\tt N}\\otimes d{\\tt N},\\\\ \n\\Delta {\\tt N}&=0.\n\\end{align}\n\n\\noindent It is apparent from here that ${\\tt Ric}\\geq 0$, which is a quite central property. Consider now a congruence of geodesics (or geodesic segments) ${\\mathcal{F}}$ for the metric ${\\tt g}$ minimizing the distance from any of their points to a (hyper)-surface ${\\mathcal{S}}$. Thus any geodesic in ${\\mathcal{F}}$ has an initial point in ${\\mathcal{S}}$. We will assume the geodesics (or geodesic segments) are inextensible beyond their last point or that the last point is the point on $\\gamma$ where $\\gamma$ stops to be length minimizing to ${\\mathcal{S}}$. It can be that such last point does not exists in which case the geodesic ``ends\" at ``infinity\".   It is known that the {\\it Cut locus} ${\\mathcal{C}}$, namely the set of last points of the geodesics in the congruence is a closed set of measure zero. Outside ${\\mathcal{C}}$ the distance function to ${\\mathcal{S}}$ is a smooth function with gradient of norm one. Given a point $p$ in $\\Sigma$, we will denote by ${\\tt s}(p)$ the distance from $p$ to ${\\mathcal{S}}$. Consider now a point $p$, not in ${\\mathcal{C}}$ and not in ${\\mathcal{S}}$ and around it consider the smooth surface formed by the set of points which have the same distance to ${\\mathcal{S}}$ than $p$ (the equidistant surface or the level set of the distance function). The second fundamental form of such surface in the outgoing direction (from ${\\mathcal{S}}$) at $p$ will be denoted by $\\Theta(p)$ or simply $\\Theta$. The mean curvature will be $\\theta(p)=tr_{h(p)}\\Theta(p)$ where $tr_{h(p)}\\Theta(p)$ means the trace of $\\Theta(p)$ with respect to the induced two-metric in the surface or level set. Thus we can think $\\theta$ as a function along geodesics $\\gamma$ in ${\\mathcal{F}}$. The mean curvature satisfies the important {\\it focussing equation} or {\\it Riccati equation} along the geodesics $\\gamma$        \n\\begin{equation}\\label{FOCEQ}\n\\theta '=-|\\Theta|^{2}-{\\tt Ric}(\\gamma',\\gamma')=-\\frac{\\theta^{2}}{2}-{\\tt Ric}(\\gamma',\\gamma')-|\\hat{\\Theta}|^{2}.\n\\end{equation}\n\n\\noindent Above, $'$ denotes derivative with respect to arc length and $\\hat{\\Theta}$ is the traceless part of $\\Theta$. Recall that\n\\begin{displaymath}\n\\Delta {\\tt s}=\\theta.\n\\end{displaymath}\n\n\\noindent Thus any estimate on $\\theta$ obtained out of the focussing equation serves as an estimate on the Laplacian of the distance function. \n\nFor instance if ${\\tt Ric}\\geq 0$ then standard estimates in comparison theory follow by discarding the last two terms in equation (\\ref{FOCEQ}) and integrating the inequality $\\theta'\\leq -\\theta^{2}\/2$. If ${\\tt s}$ is the distance function to a point, or, the same, the distance function to the boundary of a small geodesic ball plus the radius of the ball, one gets (Calabi 1958), $\\theta\\leq 2\/{\\tt s}$ and \n\\begin{displaymath}\n \\Delta {\\tt s}\\leq 2\/{\\tt s},\n\\end{displaymath} \n \n\\noindent  everywhere and in the barer sense (~\\cite{MR2243772}, pg. 262). Comparison estimates on areas and volumes of geodesic balls are obtained from \n\\begin{displaymath}\n\\frac{dA'}{dA}=\\theta,\\ \\ dV'=dA.\n\\end{displaymath}\n\n\\noindent where $dA$ is the element of area of the equidistant surfaces to ${\\mathcal{S}}$ and $dV$ is the element  of volume enclosed by $dA$. \n\nThe situation we face is similar in that the Ricci curvature is non-negative, but this time the structure of the Ricci curvature is explicitly given. By incorporating ${\\tt Ric}$ as part of the focussing inequality, namely considering\n\\begin{displaymath}\n\\theta'\\leq -\\frac{\\theta^{2}}{2}-2{\\tt N}'^{2},\n\\end{displaymath}\n\n\\noindent we will obtain a sharp estimate for $\\theta$. We will show that for any real number $a$ the quantity \n\\begin{displaymath}\n{\\mathcal{M}}_{a}=(\\frac{\\theta}{2}({\\tt s}+a)^{2}-({\\tt s}+a))N^{2},\n\\end{displaymath}\n\n\\noindent is monotonically decreasing (Proposition \\ref{MMP}) along any geodesic of the congruence and is stationary if and only if the geometry along the geodesic is of Schwarzschild form (Proposition \\ref{COM}). Thus we get the estimate\n\\begin{displaymath}\n\\theta\\leq \\frac{2}{{\\tt s}+a}(1+\\frac{{\\mathcal{M}}_{0}}{({\\tt s}+a)^{2}N^{2}}).\n\\end{displaymath}      \n\n\\noindent where ${\\mathcal{M}}_{0}$ is the value of ${\\mathcal{M}}_{a}$ at the start, on ${\\mathcal{S}}$, of the geodesic. The fundamental set of equations out of which comparison estimates can be obtained is therefore\n\\begin{align}\n&\\theta\\leq \\frac{2}{{\\tt s}+a}(1+\\frac{{\\mathcal{M}}_{0}}{({\\tt s}+a)^{2}N^{2}}),\\label{FE1}\\\\\n&\\Delta{\\tt s}=\\theta,\\ \\frac{dA'}{dA}=\\theta,\\ dV'=dA,\\label{FE2}\\\\\n&\\Delta \\ln N=0\\label{FE3}.\n\\end{align}\n\n\\noindent To use these set of equations efficiently one must first use the system\n\\begin{align*}\n&\\Delta{\\tt s}\\leq \\frac{2}{{\\tt s}+a}(1+\\frac{{\\mathcal{M}}_{0}}{({\\tt s}+a)^{2}N^{2}}),\\\\\n&\\Delta \\ln N=0,\n\\end{align*}\n\n\\noindent together with additional boundary data on $N$ and ${\\mathcal{M}}_{0}$. For the case of the application to the uniqueness of the Schwarszchild solutions, that we carry out later, the substantial information that is extracted out of this system is, in a sense, concentrated in Theorem \\ref{DC}, where a distance comparison result is established between ${\\tt s}$ and $\\hat{\\st}=2mN^{2}\/(1-N^{2})$. \n\nFrom the point of view of areas and volumes comparisons, we note that, by using equations (\\ref{FE1})-(\\ref{FE3}), the expression \n\\begin{displaymath}\n\\frac{dA}{dA_{0}}exp(\\int_{{\\tt s}_{0}}^{{\\tt s}}\\frac{2}{\\bar{{\\tt s}}+a}(1+\\frac{{\\mathcal{M}}_{0}}{\\bar{{\\tt s}}+a)^{2}N^{2}})d\\bar{{\\tt s}}),\n\\end{displaymath}\n\n\\noindent is seen to be monotonically decreasing too. From it and $dV'=dA$ suitable information on the growth of areas and volumes of geodesic balls (with center ${\\mathcal{S}}$) can be obtained. These type of estimates will play an important role in the proof of the uniqueness of the Schwarzschild solutions in Section \\ref{AVC}. \n\nYet, the structure of the harmonic-map representation of the Einstein equations is richer than the information contained in the system (\\ref{FE1})-(\\ref{FE3}). Indeed, Weitzenb\\\"och's formula for the static equations \n\\begin{displaymath}\n\\frac{1}{2}\\Delta |\\nabla f|^{2}=|\\nabla\\nabla f|^{2}+<\\nabla \\Delta f,\\nabla f>+2<\\frac{\\nabla N}{N},\\nabla f>^{2},\n\\end{displaymath}\n\n\\noindent valid for any function $f$, together with equation (\\ref{FE3}) can provide useful estimates on functions of the form $f=f(N)$. They, in turn, provide useful information on $N$. These estimates, is worth remarking, have nothing to do with the distance function. The most obvious consequence of Weitzenb\\\"ock's formula comes out when we chose $f=\\ln N$. In this case we obtain\n\\begin{displaymath}\n\\frac{1}{2}\\Delta |\\nabla \\ln N|^{2}=|\\nabla\\nabla \\ln N|^{2}+2|\\nabla \\ln N|^{2}.\n\\end{displaymath}\n\n\\noindent In applications to the uniqueness of the Scharzschild solutions, we will use however the Weintzenb\\\"ock formula with the choice $f=\\hat{\\st}=2mN^{2}\/(1-N^{2})$. This will provide the important estimate $|\\nabla \\hat{\\st}|\\leq 1$ in Section \\ref{FPCDL}, which, as we will see, it is necessary to close up the proof of the uniqueness of the Schwarzschild solutions. \n\nIt is worth remarking at this point that many of the techniques here developed carry over the much bigger family of metrics and potentials satisfying \n\\begin{align*}\n&{\\tt Ric}\\geq 2d{\\tt N}\\otimes d{\\tt N},\\\\\n&\\Delta {\\tt N}\\geq 0,\\ \\ 0<N<1.\n\\end{align*}\n\nTo show the applicability of equations (\\ref{FE1})-(\\ref{FE2}), as we said before, we will fully analyze from this perspective asymptotically flat static solutions with regular and connected horizons and recover Theorem \\ref{TRS}. It is worth remarking the naturalness from which the uniqueness of the Schwarzschild solutions will come out of these comparison techniques. Despite of that, the required analysis will be somewhat extensive. \nTo prove Theorem \\ref{TRS} we carefully compare the distance function to the horizon, ${\\tt s}$, to the function $\\hat{\\st}$, through the set of equations (\\ref{FE1})-(\\ref{FE2}). The final goal to achieve is to show the equality ${\\tt s}=\\hat{{\\tt s}}=2mN^{2}\/(1-N^{2})$ from which it will follow that ${\\mathcal{M}}_{2m}$ has to be stationary along any length-minimizing geodesic to $H$ (in this case the integral lines of $\\nabla \\hat{\\st}$) and equal to $m$. It then follows, from the sharpness of the monotonicity of ${\\mathcal{M}}$ that the solution has to be a Schwazschild solution (of positive mass). Note a technical aspect however. As $N=0$ over the horizon $H$, the metric ${\\tt g}=N^{2}g$ is singular there. Although this will make the analysis technically delicate, a satisfactory remedy is found if one replaces $H$ by a sequence $H_{\\Gamma_{i}}=\\{N=\\Gamma_{i}\\}$ of the $\\Gamma_{i}$-level set of $N$ ($\\Gamma_{i}\\downarrow 0$), approaching $H$, and perform then a limit analysis. This circumvention of the singularity at the horizon will appear often in the reasonings.   \n\nIt is worth noting that, at the moment, we do not know how to obtain Theorem \\ref{TRS}, when the horizon is not connected. The exact reproduction of the arguments that leads to the proof of Theorem \\ref{TRS} for connected horizons, applied to the case of non-connected horizons, give interesting results, which are not difficult to obtain but that will not be given here.    \n\n\nWe will now give guidelines of the structure of the article. In Section \\ref{CON2} we introduce and discuss the main monotonic quantity ${\\mathcal{M}}$, give explicit examples of the monotonicity and discuss the stationary case. This section is the core of the article. The other sections discuss further properties of ${\\mathcal{M}}$ and applications. In particular in Section \\ref{CON3} we start the discussion of asymptotically flat solutions with regular and connected horizons. In Section \\ref{SMRH} we study ${\\mathcal{M}}$ over regular horizons. In Section \\ref{SAF} we recall the notion of asymptotic flatness and cite a classical result \\cite{MR608121} on the possibility to chose special coordinates at infinity in static solutions displaying precisely the Scwarzschild-type of fall off. The existence of such coordinate system $\\{\\bar{x}\\}$ will be central. In Section \\ref{SCDL} we introduce the important notion of {\\it coordinate-distance} lag, measuring a mismatch between the distance from a point $p$ to the horizon, ${\\tt s}(p)$, and the  coordinate distance $|\\bar{x}(p)|$. In Section \\ref{SDC} we discuss our first substantial result. We prove a {\\it distance comparison} result (Theorem \\ref{DC}) between ${\\tt s}$ and $\\hat{{\\tt s}}=2m N^{2}\/(1-N^{2})$. To achieve it, we must show first that the inequality \n\\begin{displaymath}\n\\Delta{\\tt s}\\leq \\frac{2}{{\\tt s}+a}(1+\\frac{{\\mathcal{M}}_{0}}{({\\tt s}+a)^{2}N^{2}}),\n\\end{displaymath}\n\n\\noindent holds in a barer sense all over the manifold $\\Sigma$. This is done in Proposition \\ref{LSH}. Without that tool, the comparison result would not be possible to achieve. Using that we show in Section \\ref{SPI} that the Penrose inequality $A\\leq 16\\pi m^{2}$, where $A$ is the area of the connected horizon and $m$ is the ADM mass, must hold. In Section \\ref{OSPI} we show, again using the distance comparison,  that the opposite Penrose inequality must hold, namely $A\\geq 16\\pi m^{2}$. Thus, after Section \\ref{OSPI} we would have proved that $A=16\\pi m^{2}$. Despite the strong implications of this inequality, the uniqueness of the Schwarzschid solutions requires further analysis. This is carried out in Section \\ref{USS}. Indeed it is in this section that it is proved that ${\\tt s}=\\hat{{\\tt s}}$. This follows from a further study of the coordinate-distance lag in Section \\ref{FPCDL} where it is shown that it must be zero. An elaboration of the area and volume comparison in Section \\ref{AVC} finishes the analysis of all the elements of the proof which is summarized in Theorem \\ref{TUSFIN}.  Further explanations on the contents and strategies are given at the beginning of each Section. \n\nWe will use alternatively the notation $(\\Sigma,g,N)$ or $(\\Sigma,{\\tt g},{\\tt N})=(\\Sigma,{\\tt g},\\ln N)$, used according to which representation is best suited to describe a claim or a statement. When we say that $(\\Sigma, {\\tt g}, \\ln N)$ is an asymptotically flat static solution with regular and connected horizon, we mean that $(\\Sigma,g,N)$ is an asymptotically flat static solution with regular horizon as was described before, but that we will working in its harmonic map representation.   \n\n\\section{A comparison approach to static solutions in the harmonic map representation.}\\label{CON2}\n\nLet $(\\Sigma,{\\tt g},\\ln N)$ be a static solution in the harmonic representation. To every oriented integrable congruence ${\\mathcal{F}}$ of ${\\tt g}$-geodesics, we will associate a family of real functions $\\{{\\mathcal{M}}_{a},a\\in \\field{R}\\}$ defined over the range of ${\\mathcal{F}}$. We will show that, fixed $a$, ${\\mathcal{M}}_{a}(\\gamma ({\\tt s}))$ is monotonically decreasing for any $\\gamma\\in {\\mathcal{F}}$ (${\\tt s}$ is the ${\\tt g}$-arc length, increasing in the positive direction). This central fact will follow by making use of the focusing equation (\\ref{FOCEQ}). The definition of ${\\mathcal{M}}_{a}$ and the proof of its monotonicity are given in the Proposition below. To avoid excessive notation we will use the following convention in the notation: for every function $f$ defined over the range of ${\\mathcal{F}}$ (for example $f=\\theta$ or $f=N$) and $\\gamma\\in {\\mathcal{F}}$ we will write $f:=f(\\gamma(s))$ and $df(\\gamma(s))\/ds:=df\/ds:=f'$. Also, for the same reason of economy and simplicity, we will suppress the sub-index $a$ and write simply ${\\mathcal{M}}$.  \n\n\\begin{Proposition}\\label{MMP}\nLet ${\\mathcal{F}}$ be an oriented integrable congruence of geodesics. Let $\\gamma({\\tt s})$, ${\\tt s}\\in [{\\tt s}_{0},{\\tt s}_{1}]$ be a geodesic in ${\\mathcal{F}}$. Let $a$ be a real number and let ${\\tilde{\\tt s}}=a+{\\tt s}$. Then we have\n\\begin{equation}\\label{MME}\n((\\frac{\\theta}{2}{\\tilde{\\tt s}}^{2}-{\\tilde{\\tt s}})N^{2})'=-{\\tilde{\\tt s}}^{2}N^{2}\\frac{|\\hat{\\Theta}|^{2}}{2}-({\\tilde{\\tt s}}\\frac{\\theta}{2}-1 -{\\tilde{\\tt s}}\\frac{N'}{N})^{2}N^{2}.\n\\end{equation}\n\n\\noindent Therefore, fixed any real number $a$, the quantity ${\\mathcal{M}}=(\\frac{\\theta}{2}{\\tilde{\\tt s}}^{2}-{\\tilde{\\tt s}})N^{2}$ is monotonically decreasing along any $\\gamma\\in {\\mathcal{F}}$ (the notation ${\\mathcal{M}}$ accounts for ``{\\it mass}''). \n\\end{Proposition}\n \n\\noindent {\\it Proof:}\n\n\\vs\nWe compute\n\\begin{displaymath}\n((\\frac{\\theta}{2}\\tilde{{\\tt s}}^{2}-\\tilde{{\\tt s}})N^{2})'=\\frac{\\theta'}{2}\\tilde{{\\tt s}}^{2}N^{2}+\\theta \\tilde{{\\tt s}}N^{2}-N^{2}+\\theta\\tilde{{\\tt s}}^{2}N N'-\\tilde{{\\tt s}}2N N'.\n\\end{displaymath}   \n \n\\noindent We use now the focusing equation (\\ref{FOCEQ}) to get\n\\begin{displaymath}\n((\\frac{\\theta}{2}\\tilde{{\\tt s}}^{2}-\\tilde{{\\tt s}})N^{2})'=-\\frac{|\\hat{\\Theta}|^{2}}{2}\\tilde{{\\tt s}}^{2} N^{2} - \\frac{\\theta^{2}}{4}\\tilde{{\\tt s}}^{2} N^{2} -\\tilde{{\\tt s}} N'^{2} + \\theta \\tilde{{\\tt s}}N^{2}-N^{2}+\\theta\\tilde{{\\tt s}}^{2}N N'-\\tilde{{\\tt s}}2N N'.\n\\end{displaymath}    \n \n\\noindent The six terms following the first on the right hand side of this expression can be arranged as $-(\\tilde{{\\tt s}}\\theta\/2-1-N'\/N)^{2}N^{2}$, thus obtaining (\\ref{MME}).\\hspace{\\stretch{1}}$\\Box$\n \n\\begin{Example} {\\rm ({\\it The Schwarzschild case.})\\label{EXA1} Consider a Schwarzschild metric of mass $m$, of arbitrary sign, in the presentations, according to the sign of the mass, of equations (\\ref{SchP}) or (\\ref{SchP2}). Note that ${\\tt g}=dr^{2}+r^{2}(1-2m\/r)d\\Omega^{2}$ and $N^{2}=(1-2m\/r)$. For any given point $q$ in $S^{2}$ consider the ray $[2m,\\infty)\\times \\{q\\}$ (if $m\\geq 0$) or $(o,\\infty)\\times \\{q\\}$ (if $m<0$) parameterized by the arc length ${\\tt s}=r-2m$ of ${\\tt s}=r$ (respectively to the sign of the mass). In either case we compute the mean curvature $\\theta$ as\n\\begin{equation}\n\\theta=\\frac{2}{r}+\\frac{2m}{r(r-2m)}=\\frac{2}{r}\\frac{(r-m)}{r-2m}.\n\\end{equation}\n\n\\noindent Let $b=a-2m$ if $m\\geq 0$ and $b=a$ if $m<0$, then the quantity ${\\mathcal{M}}$ has the following form,\n\\begin{displaymath}\n{\\mathcal{M}}=(\\frac{1}{r}\\frac{r-m}{r-2m}(r+b)^{2}-(r+b))(1-\\frac{2m}{r})=((m+b)-\\frac{m b}{r})(1+\\frac{b}{r}).\n\\end{displaymath}\n\n\\noindent independently of the sign of the mass. Taking the derivative with respect to arc length and rearranging terms we obtain \n\\begin{displaymath}\n{\\mathcal{M}}'=\\frac{b^{2}}{r}(\\frac{2m}{r}-1),\n\\end{displaymath}\n\n\\noindent which is explicitly non-positive independently of the sign of the mass. This shows the monotonicity of ${\\mathcal{M}}$ for any value of $b$. Note that $lim_{{\\tt s} \\rightarrow \\infty}{\\mathcal{M}}=m+b$. Observe too that when $b=0$, i.e. $a=2m$, then ${\\mathcal{M}}$ is constant and equal to $m$.}\\hspace{\\stretch{1}}$\\Box$ \n\\end{Example}\n\n\\begin{Example} {\\rm ({\\it The flat solutions.}) For a flat solution $(\\Sigma_{F},g_{F},N_{F})$ we have ${\\tt g}=r^{2}dr^{2} +r^{2}h_{F}$. Making $r^{2}\/2=s$ we get ${\\tt g}=ds^{2}+2sh_{F}$ and $N^{2}=2s$ where $s>0$. For any point $p$ in $T^{2}$ consider the ray $[0,\\infty)\\times {p}$. Consider the congruence of geodesics conformed by all these rays. The mean curvature is calculated as $\\theta=1\/s$. Thus for any real number $a$ we have \n\\begin{equation}\n{\\mathcal{M}}=(\\frac{1}{2s}(s+a)^{2}-(s+a))2s=-s^{2}+a^{2},\n\\end{equation}\n\n\\noindent which is monotonically decreasing in the domain of $s$, namely $(0,\\infty)$. \n\nNote that for the ``dual'' solution $(\\Sigma_{F},g_{F},1\/N_{F})$ we have, for any real number $a$, the expression ${\\mathcal{M}}=-1\/4 +a^{2}\/s^{2}$ which is monotonically decreasing in the domain of $s$, namely $(0,\\infty)$. Note that when $a=0$ then ${\\mathcal{M}}$ is stationary and equal to $-1\/4$.}\\hspace{\\stretch{1}}$\\Box$\n\\end{Example}\n\n\\vs\nThe next proposition discusses the case when ${\\mathcal{M}}_{a}$ is stationary.\n\n\\begin{Proposition}\\label{COM}\nLet ${\\mathcal{F}}$ be an oriented and integrable congruence of geodesics. When, for a given $a$, ${\\mathcal{M}}$ is constant along a geodesic segment $\\gamma({\\tt s})$, ${\\tt s}\\in [{\\tt s}_{1},{\\tt s}_{2}]$ then along $\\gamma$ we have\n\\begin{equation}\\label{SM1}\n\\hat{\\Theta}=0,\n\\end{equation}\n\\noindent and\n\\begin{equation}\\label{SM2}\nN^{2}=N_{0}^{2}+2\\frac{{\\mathcal{M}}_{0}}{{\\tt s}_{0}+a}-2\\frac{{\\mathcal{M}}_{0}}{{\\tt s}+a},\n\\end{equation}\n\n\\noindent where $N_{0}$ and ${\\mathcal{M}}_{0}$ are the values of $N$ and ${\\mathcal{M}}$ at ${\\tt s}={\\tt s}_{0}\\in ({\\tt s}_{1},{\\tt s}_{2})$. We also obtain\n\\begin{equation}\\label{SM3}\n\\theta=\\frac{2}{{\\tt s}+a}+2\\frac{{\\mathcal{M}}_{0}}{({\\tt s}+a)^{2}N^{2}}.\n\\end{equation}   \n\\end{Proposition}\n\n\\noindent {\\it Proof:} \n\n\\vs\nIf along a geodesic $\\gamma$ the value of ${\\mathcal{M}}$ remains constant, then the right hand side of (\\ref{MME}) must be identically zero. This implies that \n\\begin{displaymath}\n\\hat{\\Theta}=0,\n\\end{displaymath}\n\\noindent which shows (\\ref{SM1}), and also implies that \n\\begin{displaymath}\n\\tilde{{\\tt s}}\\frac{\\theta}{2}-1-\\tilde{{\\tt s}}\\frac{N'}{N}=0.\n\\end{displaymath}\n\n\\noindent Multiply now this expression by $\\tilde{{\\tt s}}$ and rearrange it as\n\\begin{equation}\\label{TEM}\n\\tilde{{\\tt s}}^{2}\\frac{\\theta}{2}-\\tilde{{\\tt s}}=\\tilde{{\\tt s}}^{2}\\frac{N'}{N}.\n\\end{equation}\n\n\\noindent Recall that ${\\mathcal{M}}=(\\tilde{{\\tt s}}^{2}\\frac{\\theta}{2}-\\tilde{{\\tt s}})N^{2}$. Using this expression, the equation (\\ref{TEM}), and (because we are assuming that ${\\mathcal{M}}$ is constant) writing ${\\mathcal{M}}={\\mathcal{M}}_{0}={\\mathcal{M}}({\\tt s}_{0})$, we obtain \n\\begin{displaymath}\n{\\mathcal{M}}_{0} =\\tilde{{\\tt s}}^{2}N N'=\\tilde{{\\tt s}}^{2} \\frac{(N^{2})'}{2}.\n\\end{displaymath}\n\n\\noindent Moving $\\tilde{{\\tt s}}^{2}$ to the denominator of the left hand side and integrating (in ${\\tt s}$) from ${\\tt s}={\\tt s}_{0}$ to ${\\tt s}$ we obtain (\\ref{SM2}). To obtain (\\ref{SM3}) solve for $\\theta$ in $(\\tilde{{\\tt s}}^{2}\\frac{\\theta}{2}-\\tilde{{\\tt s}})N^{2}={\\mathcal{M}}_{0}$.\n\\hspace{\\stretch{1}}$\\Box$ \n\n\\begin{Remark} {\\rm (Further remarks to Proposition \\ref{COM})\nObserve form Proposition (\\ref{COM}) that (if for some number $a$) ${\\mathcal{M}}$ is constant along a geodesic $\\gamma$ of infinite length and $lim_{{\\tt s}\\rightarrow \\infty} N(\\gamma({\\tt s}))=1$, then making the change of variables $r={\\tt s}+a$ in (\\ref{SM2}) and (\\ref{SM3}) we obtain, along $\\gamma$, the expressions\n\\begin{displaymath}\nN^{2}(r)=1-\\frac{2{\\mathcal{M}}_{0}}{r},\n\\end{displaymath}\n\\begin{displaymath}\n\\theta=\\frac{2}{r}+\\frac{2{\\mathcal{M}}_{0}}{r(r-2{\\mathcal{M}}_{0})}=\\frac{2}{r}\\frac{(r-{\\mathcal{M}}_{0})}{r-2{\\mathcal{M}}_{0}}.\n\\end{displaymath}\n\n\\noindent and, including (\\ref{SM1})\n\\begin{displaymath}\n\\hat{\\Theta}=0,\n\\end{displaymath}\n\n\\noindent which, comparing with Example \\ref{EXA1}, are exactly of Schwarzschild form if we identify ${\\mathcal{M}}_{0}$ with ``a\" ADM mass $m$. Moreover if $\\gamma$ is defined on $({\\tt s}_{0}=0,\\infty)$ and $\\gamma({\\tt s}_{0})$ ``lies'' on a ``horizon'' ($\\lim_{{\\tt s}\\rightarrow 0} N(\\gamma({\\tt s}))=0$), then $N_{0}=0$ and $1=N_{0}^{2}+2{\\mathcal{M}}_{0}\/({\\tt s}_{0}+a)=2m\/a$. Therefore $a=2m$ and ${\\tt s}=r-2m$. Note that $m={\\mathcal{M}}_{0}$ cannot be negative otherwise $\\theta$ reaches infinity for an ${\\tt s}\\in (0,\\infty)$ (thus the whole $\\gamma$ cannot belong to ${\\mathcal{F}}$). Thus we establish the same relation ${\\tt s}=r-2m$ as in a Schwarzschild solution of positive mass. \nOn the other hand if $\\gamma$ is defined on $({\\tt s}_{0}=0,\\infty)$ and $\\gamma({\\tt s}_{0})$ ``lies'' on a ``naked singularity'' ($\\lim_{{\\tt s}\\rightarrow 0}N(\\gamma({\\tt s}))=+\\infty$), then $a=0$ and ${\\tt s}=r$. Note that in this case $m={\\mathcal{M}}_{0}$ must be negative otherwise $N^{2}=1-2m\/r=1-2m\/{\\tt s}$ gets negative for small ${\\tt s}>0$. Thus we establish the same relation ${\\tt s}=r$ as in a Schwarzschild solution of negative mass.}\n\\end{Remark}\n\n\n\\begin{Remark}\nThere are several ways to include the summand $-2(\\dot{N}\/N)^{2}$ to obtain an estimation on the growth of $\\theta$. The following Proposition, whose proof is left to the reader, is one such instance. Although we will not use it for the rest of the article, it illustrates very well, the many ways in which the focussing equation can be used to extract geometric information. \n\n\\begin{Proposition} Let $\\theta$ be the mean curvature of the integrable congruence ${\\mathcal{F}}$. Let $\\gamma({\\tt s})$, ${\\tt s}\\in [{\\tt s}_{0},{\\tt s}_{1}]$ be in ${\\mathcal{F}}$. Then we have\n\n\\begin{enumerate}\n\\item $\\theta N^{2}$ is monotonically decreasing, namely $(\\theta N^{2})\\dot{}\\leq -(\\frac{\\theta N}{\\sqrt{2}}-\\sqrt{2}\\dot{N})^{2}$. Therefore we have $\\theta\\leq \\theta_{0}(N_{0}\/N)^{2}$, where $\\theta_{0}=\\theta({\\tt s}_{0})$ and $N_{0}=N({\\tt s}_{0})$. \n\n\\item Suppose that $\\theta({\\tt s})>0$ for all ${\\tt s}$ in $[{\\tt s}_{0},{\\tt s}_{1}]$. Then we have\n\\begin{equation}\\label{FE}\n\\theta({\\tt s})\\leq \\frac{1}{\\frac{1}{\\theta_{0}}+\\frac{{\\tt s}-{\\tt s}_{0}}{2}+\\frac{1}{2\\theta_{0}^{2}N_{0}^{4}}\\frac{(N^{2}-N_{0}^{2})^{2}}{({\\tt s}-{\\tt s}_{0})}}.\n\\end{equation}\n\n\\noindent As $\\theta$ is monotonically decreasing the same formula holds for all ${\\tt s}$ in the domain where ${\\tt \\gamma}$ is length minimizing provided only $\\theta_{0}>0$. \n\n\\end{enumerate}\\hspace{\\stretch{1}}$\\Box$\n\\end{Proposition}\n\nEquation (\\ref{FE}) clearly displays the influence of the Lapse $N$ in the focussing of geodesics beyond the natural focussing that comes out of the non-negativity of the Ricci curvature. Equation (\\ref{FE}) can serve, in particular, to obtain information on the relationship between volume growth of tubular neighborhoods of a horizon and the growth of $N$ from it. \n\\end{Remark}\n\n\\section{Applications to asymptotically flat static solutions with regular and connected horizons.} \\label{CON3}\n\nIn this section we show that any asymptotically flat static solution with regular and connected horizon must satisfy the Penrose inequality. This is proved in Section \\ref{SPI}. Separately, in Section \\ref{OSPI} we will prove that one such solution must satisfy the opposite Penrose inequality and that the horizon must be geometrically round. This will lead us into the verge of proving Theorem \\ref{TRS} which is carried out in Section \\ref{USS}. To achieve the inequalities some preliminary material is introduced in Sections \\ref{SMRH}, \\ref{SAF}, \\ref{SCDL} and \\ref{SDC}. In Section \\ref{SMRH} we compute ``the value of ${\\mathcal{M}}$'' for the ``congruence of geodesics emanating perpendicularly to $H$'' (note that ${\\tt g}$ is singular on $H$) which we will be usied crucially in the other Sections. Technically we will elude the fact that ${\\tt g}$ is singular on $H$ by considering instead of $H$ suitable sequences $\\{H_{\\Gamma_{i}}\\}$ of two-surfaces approaching $H$ as $i\\rightarrow \\infty$. In this way ``the value of ${\\mathcal{M}}$ over $H$'' will be defined as a limit. Similarly we will define ${\\tt s}(p):=dist_{{\\tt g}}(p,H):=\\lim_{i\\rightarrow \\infty} dist_{{\\tt g}}(p,H_{\\Gamma_{i}})$.  In Section \\ref{SMRH} we recall the notion of {\\it Asymptotic Flatness} and introduce, following \\cite{MR608121}, a coordinate system adapted to asymptotically flat static solutions that will be very useful later. In Section \\ref{SCDL} we introduce the notion of {\\it Coordinate-Distance Lag} which is necessary to prove, in Theorem \\ref{DC}  of Section \\ref{SDC},  a central {\\it Distance Comparison}  where we establish a lower bound for the ${\\tt g}$-distance function to the horizon (the function ${\\tt s}$) in terms of a certain function of $N$, $m$ and $A$ (the function $\\hat{\\hat{{\\tt s}}}$). For any divergent sequence of points $\\{p_{i}\\}$ the coordinate-distance lag associated to $\\{p_{i}\\}$ is defined as $\\bar{\\delta}(\\{p_{i}\\})=\\limsup {\\tt s}(p_{i})-r(p_{i})+2m$, where $r=|\\bar{x}|$ and $\\{\\bar{x}=(x_{1},x_{2},x_{3})\\}$ is the coordinate system introduced in Section \\ref{SAF} and it will be seen to be $\\bar{\\delta}(\\{p_{i}\\})=\\limsup {\\tt s}(p_{i})-\\hat{\\st}(p_{i})$. The Penrose inequality in Section \\ref{SPI} is then proved by  showing first, using a standard comparison of mean curvatures, that if $P:=A\/(16\\pi m)>1$ (i.e. the Penrose inequality does not hold) then there is a divergent sequence whose coordinate-distance lag is non-negative  (Corollary \\ref{CDC}) and on the other hand proving, using the distance comparison of Section \\ref{SDC}, that if $P>1$ then the coordinate-distance lag must be negative for any divergent sequence (Proposition \\ref{PDC}). This reaches a contradiction. To prove the opposite Penrose inequality it is shown that the Gaussian curvature $\\kappa$ of $H$ must satisfy $\\kappa\\geq 4(4\\pi m\/A)^{2}$ to prevent a violation of the distance comparison near the horizon. integrating this inequality over $H$ and using Gauss-Bonnet the opposite Penrose inequality is achieved. As a byproduct of both inequalities one obtains that the horizon must be geometrically round, namely that $\\kappa=4\\pi\/A$.  \n\n\\subsection{The value of ${\\mathcal{M}}$ over regular horizons.}\\label{SMRH}\n\nLet $(\\Sigma,{\\tt g},\\ln N)$ be an static solution and let $H$ be a regular and connected horizon. Consider an embedded (orientable) surface ${\\mathcal{S}}\\subset \\Sigma\\setminus H$. Let $n_{1}$ and $n_{2}$ be the two unit-normal vector fields to ${\\mathcal{S}}$. As we noted before if ${\\mathcal{F}}$ is the congruence of geodesics emanating perpendicularly to ${\\mathcal{S}}$ and following one of the perpendicular directions to ${\\mathcal{S}}$, say $n_{1}$, then the mean curvature $\\theta$ of the congruence ${\\mathcal{F}}$ over ${\\mathcal{S}}$ is equal to the mean curvature of the surface ${\\mathcal{S}}$ in the direction of $n_{1}$. Now to define ${\\mathcal{M}}$ over $H$ (where ${\\tt g}$ is singular) for the ``congruence of geodesics emanating perpendicular to $H$'' we will calculate ${\\mathcal{M}}$ over a suitable sequence of surfaces and then take the limit as the surfaces approache $H$. Such calculation is performed in the paragraphs below. The following Notation will be used in this Section and those that follow.\n\n\\begin{Notation}\\label{NOT1}\nLet $\\Gamma_{0}$ be a number sufficiently small in such a way that for any $\\Gamma\\leq \\Gamma_{0}$, $\\Gamma$ is a regular value for the lapse $N$ and the set $H_{\\Gamma}:=\\{N=\\Gamma\\}$ is isotopic to $H$ (note that $|\\nabla N|\\neq 0$ over a regular horizon $H$). One such $\\Gamma_{0}$ will be called regular. For any two $\\Gamma<\\bar{\\Gamma}$ denote by $\\Omega_{\\Gamma,\\bar{\\Gamma}}$ the closed region enclosed by $H_{\\Gamma}$ and $H_{\\bar{\\Gamma}}$. The region enclosed by $H_{\\Gamma}$ and $H$ will be denoted by $\\Omega_{H,H_{\\Gamma}}$. \n\\end{Notation}\n\n\\noindent Let $\\{\\Gamma_{i}\\}_{i=1}^{i=\\infty}$ be a sequence such that $\\Gamma_{i}\\downarrow 0$ and $\\Gamma_{i}\\leq \\Gamma_{0}$ with $\\Gamma_{0}$ as in Notation \\ref{NOT1}. Define \n\\begin{displaymath}\n{\\mathcal{M}}_{H}:= \\lim_{\\Gamma_{i}\\rightarrow 0} (\\frac{\\theta}{2}a^{2}-a)N^{2}|_{H_{\\Gamma_{i}}}.     \n\\end{displaymath}\n\n\\noindent The next Proposition shows the limit above exists (so it is well defined) and is always constant over $H$. Define $|\\nabla N|_{H}=|\\nabla N|_{g}|_{H}$.\n\n\\begin{Proposition}\\label{VMH}\nLet $(\\Sigma,g,N)$ be a static solution with regular horizon $\\partial \\Sigma$. Let $H$ be a connected component of $\\partial \\Sigma$. Then we have \n\\begin{equation}\\label{MH}\n{\\mathcal{M}}_{H}=|\\nabla N|_{H}a^{2}. \n\\end{equation}\n\n\\end{Proposition}\n\n\\noindent {\\it Proof:} \n\n\\vs\nDenote (as we have done before) by $\\theta$ the mean curvature of $H_{\\Gamma}$ with respect to ${\\tt g}$ and $\\theta_{g}$ the mean curvature with respect to $g$. From the conformal relation ${\\tt g}=N^{2}g$ we know that\n\\begin{displaymath}\t\n\\theta=\\frac{\\theta_{g}}{N}+2\\frac{n(N)}{N^{2}},\n\\end{displaymath}\n\n\\noindent where $n(N)$ is the normal derivative of $N$ in the outgoing direction (outgoing to $\\partial \\Omega_{H,H_{\\Gamma_{i}}}$ and $n$ a unit vector with respect to $g$). Thus we get\n\\begin{displaymath}\n(\\frac{\\theta}{2}a^{2}-a)N^{2}=a^{2}n(N)+\\frac{a^{2}\\theta_{g}N}{2}-aN^{2}.\n\\end{displaymath}\n\n\\noindent We get equation $(\\ref{MH})$ in the limit when $\\Gamma_{i}\\rightarrow 0$. \\hspace{\\stretch{1}}$\\Box$\n\n\\subsection{Asymptotically flat static solutions.}\\label{SAF}\n\nWe will use a useful characterization of asymptotically flat static solutions $(\\Sigma, {\\tt g}, \\ln N)$ due to Beig and Simon \\cite{MR608121}. Following \\cite{MR608121} we say that $(\\Sigma,{\\tt g},\\ln N)$ is {\\it asymptotically flat} iff there is a coordinate system $\\{{\\bar{x}=(x_{1},x_{2},x_{3})}$\\ {\\rm with}\\ $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=|\\bar{x}|^{2}\\geq |\\bar{x}|_{0}^{2}\\}$ outside a a compact set in $\\Sigma$ such that\n\\begin{enumerate}\n\n\\item $\\ln N=O^{2}(\\frac{1}{|\\bar{x}|})$ and $\\ {\\tt g}_{ij}-\\delta_{ij}=O^{2}(\\frac{1}{|\\bar{x}|^{2}})$; where we use the notation $\\phi(\\bar{x})=O^{2}(f(|\\bar{x}|))$ to mean that for some positive numbers $c_{1}$, $c_{2}$ and $c_{3}$ we have \n\\begin{displaymath}\n|\\phi |\\leq c_{1}|f(|\\bar{x}|)|,\\ |\\partial_{i} \\phi |\\leq c_{2} |\\partial_{|\\bar{x}|} f(|\\bar{x}|)|\\ {\\rm and}\\ |\\partial_{i}\\partial_{j} \\phi|\\leq c_{3}|\\partial^{2}_{|\\bar{x}|} f(|\\bar{x}|)|.\n\\end{displaymath}  \n\n\\item The second derivatives of $\\ln N$ and ${\\tt g}_{ij}-\\delta_{ij}$ have bounded $C^{\\alpha}$-norm (defined with respect to the coordinate system $\\{\\bar{x}\\}$) bounded; namely if $\\phi=\\partial_{k}\\partial_{l} \\ln N$ or $\\phi=\\partial_{k}\\partial_{l} ({\\tt g}_{ij}-\\delta_{ij})$ for all $1\\leq k,l,i,j\\leq 3$ then\n\\begin{displaymath}\n\\|\\phi\\|_{C^{\\alpha}}=\\sup_{|\\bar{x}-\\bar{x}'|\\leq 1}\\frac{|\\phi(\\bar{x})-\\phi(\\bar{x}')|}{|\\bar{x}-\\bar{x}'|^{\\alpha}}<\\infty.\n\\end{displaymath}   \n\n\\end{enumerate}\n\n\\begin{Proposition}\\label{PBS}{\\rm{(\\bf Beig-Simon\\ \\cite{MR608121})}} \nLet $(\\Sigma,{\\tt g},\\ln N)$ be an asymptotically flat static solution. Then, there is a coordinate system $\\{\\bar{x}=(x_{1},x_{2},x_{3}),\\ |\\bar{x}|\\geq |\\bar{x}|_{1}\\}$ (not necessarily equal to the one defining asymptotic flatness), such that\n\\begin{equation}\\label{LNE}\n\\ln N^{2}=-\\frac{2m}{|\\bar{x}|}+O^{2}(\\frac{1}{|\\bar{x}|^{3}}),\n\\end{equation}\n\\begin{equation}\n{\\tt g}_{ij}=\\delta_{ij}-\\frac{m^{2}}{|\\bar{x}|^{4}}(\\delta_{ij}|\\bar{x}|^{2}-x_{i}x_{j})+O^{2}(\\frac{1}{|\\bar{x}|^{3}}).\n\\end{equation}\n\n\\noindent where $|\\bar{x}|^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$ and $m$ is the ADM mass of the solution.\n\\end{Proposition}\n\n\\noindent Note that the remainders are $O^{2}(1\/|\\bar{x}|^{3})$ in particular $\\ln N$ has zero dipole moment. This fact will be important later. Note too that $|\\bar{x}|^{2}d\\Omega^{2}=|\\bar{x}|^{2}(d\\theta^{2}+\\sin^{2}\\theta d\\varphi^{2})=(\\delta_{ij}-(x_{i}x_{j})\/|\\bar{x}|^{2})dx_{i}dx_{j}$ therefore we have \n\\begin{displaymath}\n{\\tt g}=\\delta_{ij}dx^{i}dx^{j}-m^{2}d\\Omega^{2}+O^{2}(\\frac{1}{|\\bar{x}|^{3}})=(d|\\bar{x}|)^{2}+(|\\bar{x}|^{2}-m^{2})d\\Omega^{2}+O^{2}(\\frac{1}{|\\bar{x}|^{3}}).\n\\end{displaymath}\n\n\\noindent To make contact with the representation (\\ref{SchP}) of the Schwarzschild solution proceed as follows. Let $(|\\bar{x}|,\\theta,\\varphi)$ be the spherical coordinate system associated to the coordinate system $\\{\\bar{x}\\}$. Make the change of variables $(|\\bar{x}|,\\theta,\\varphi)\\rightarrow (r,\\theta,\\varphi)$ with $r=|\\bar{x}|+m$. Then, for the metric ${\\tt g}$, we obtain\n\\begin{displaymath}\n{\\tt g} = dr^{2}+r^{2}(1-\\frac{2m}{r})d\\Omega^{2}+O^{2}(\\frac{1}{r^{3}})={\\tt g}_{S}+O^{2}(\\frac{1}{r^{3}}).\n\\end{displaymath}\n\n\\noindent For the Lapse $N$ instead, we obtain the following expansion. From (\\ref{LNE}) we have\n\\begin{displaymath} \nN^{2}=1-\\frac{2m}{|\\bar{x}|}+\\frac{2m^{2}}{|\\bar{x}|^{2}}+O^{2}(\\frac{1}{|\\bar{x}|^{3}}).\n\\end{displaymath}\n\n\\noindent Now use \n\\begin{displaymath}\n\\frac{1}{|\\bar{x}|}=\\frac{1}{r-m}=\\frac{1}{r}+\\frac{m}{r^{2}}+\\frac{m^{2}}{r^{3}}+O^{2}(\\frac{1}{r^{4}}).\n\\end{displaymath}\n\n\\noindent to get\n\\begin{displaymath}\nN^{2}=1-\\frac{2m}{r}+O^{2}(\\frac{1}{r^{3}}).\n\\end{displaymath}\n\nWe can thus rephrase the Proposition \\ref{PBS} in the following form \n\n\\begin{Proposition}\\label{PBS2} \nLet $(\\Sigma,{\\tt g},\\ln N)$ be an asymptotically flat static solution. Then, there is a coordinate system $\\{\\bar{x}=(x_{1},x_{2},x_{3}),\\ (x_{1}^{2}+x_{2}^{2}+x_{3}^{2})^{\\frac{1}{2}}=r\\geq r_{1}\\}$ (not necessarily equal to the one defining asymptotic flatness), such that\n\\begin{equation}\\label{LNE2}\nN^{2}=1-\\frac{2m}{r}+O^{2}(\\frac{1}{r^{3}}),\n\\end{equation}\n\\begin{equation}\n{\\tt g}=dr^{2}+r^{2}(1-\\frac{2m}{r})d\\Omega^{2}+O^{2}(\\frac{1}{r^{3}}).\n\\end{equation}\n\n\\noindent where $m$ is the ADM mass of the solution.\n\\end{Proposition}\n\nThe following Proposition on the asymptotic of the mean curvatures of the coordinate spheres $S_{r}=\\{p\/r(p)=r\\}$ is now direct.\n\n\\begin{Proposition}\\label{PMC}\nLet $(\\Sigma,g,N)$ be an asymptotically flat static solution and consider a coordinate system as in Proposition \\ref{PBS2}. Then, the mean curvature $\\theta_{r}$ of the level surfaces $S_{r}=\\{p\/r(p)=r\\}$ satisfy, at every point in $S_{r}$, the estimate\n\\begin{equation}\n\\theta_{r}=\\frac{2}{r}+\\frac{2m}{r^{2}}+O(\\frac{1}{r^{3}}). \n\\end{equation}\n\\end{Proposition}\n\n\\subsection{The coordinate-distance lag.}\\label{SCDL}\n\nLet $(\\Sigma,{\\tt g}, \\ln N)$ be an asymptotically flat static solution with regular and connected horizon $H$. We would like first to introduce the {\\it distance function} to $H$, the definition of which is more or less evident. We will follow the Notation \\ref{NOT1}. \n\nLet $p\\in \\Sigma\\setminus H$ and let $\\{\\Gamma_{i}\\}_{i=1}^{i=\\infty}$ be a strictly decreasing sequence such that, $\\Gamma_{i}\\leq \\Gamma_{0}$, $\\lim \\Gamma_{i}=0$ and $p\\notin \\Omega_{H_{1},H}$. We note that if $j>i$ then \n\\begin{displaymath}\ndist(p,H_{\\Gamma_{i}})<dist(p,H_{\\Gamma_{j}}),\n\\end{displaymath}\n\n\\noindent and we have \n\\begin{equation}\\label{INEQ}\ndist(p,H_{\\Gamma_{i}})\\leq dist(p,H_{\\Gamma_{j}})\\leq dist(p,H_{\\Gamma_{i}})+diam(\\Omega_{H_{\\Gamma_{j}},H_{\\Gamma_{i}}}),\n\\end{equation}\n\n\\noindent where the diameter of $\\Omega_{H_{\\Gamma_{j}},H_{\\Gamma_{i}}}$, $diam(\\Omega_{H_{\\Gamma_{j}},H_{\\Gamma_{i}}})$, tends to zero as $i (<j)\\rightarrow \\infty$. Denote $s_{\\Gamma}(p):=dist(p, H_{\\Gamma})$. The inequality (\\ref{INEQ}) shows that \n\\begin{displaymath}\ns(p):=\\lim_{i\\rightarrow \\infty} s_{\\Gamma_{i}}(p),\n\\end{displaymath}\n\n\\noindent for any sequence $\\{\\Gamma_{i}\\}$ as above, is well defined and independent on $\\{\\Gamma_{i}\\}$. We thus define the distance from $p$ to $H$ in that way. Note that given a point $p$ in $\\Sigma\\setminus H$ one can always construct a length minimizing geodesic from $p$ to $H$ by taking the limit of length minimizing geodesics from $p$ to $H_{\\Gamma_{i}}$. This fact will be used later without further mention.\n\nNow consider the Schwazschild solution $\\bar{\\tt g}_{S}=dr^{2}+(1-\\frac{2m}{r})r^{2}d\\Omega^{2}$ and consider a ray $\\gamma (r)=(r,\\theta_{0},\\varphi_{0})$, $r\\in [2m,\\infty)$, which is, naturally, length minimizing between any two of its points. Let ${\\tt s}(\\gamma(r))$ be the length of $\\gamma$ between $r=2m$ and $r$. Then ${\\tt s}(\\gamma(r))=r-2m$ and therefore the limit $lim_{r\\rightarrow \\infty} {\\tt s}(\\gamma(r))-r+2m=0$. Now consider a ray $\\gamma(\\tau)$ on an (another) asymptotically flat static solution with regular and connected horizon $H$, joining $H$ to infinity. Then in this different scenario, instead, the limit $\\lim s(\\gamma(\\tau))-r(\\gamma(\\tau))+2m$ may be different from zero. We advocate now to define {\\it the coordinate distance lag} measuring precisely this a priori mismatch. \n\\begin{Definition} \\label{CDLD}\nLet $(\\Sigma, {\\tt g}, \\ln N)$ be an asymptotically flat static solution with connected and regular horizon. Let $\\{\\bar{x}=(x_{1},x_{2},x_{3}),\\ |\\bar{x}|=r\\geq r_{1}\\}$ be a coordinate system as in Proposition \\ref{PBS2}. Let $\\{p_{i}\\}$ be a diverging sequence of points (i.e. $s(p_{i})\\rightarrow \\infty$) (lying inside the range of $\\{\\bar{x}\\}$). Then, the coordinate distance lag, $\\bar{\\delta}$, associated to the sequence $\\{p_{i}\\}$ is defined as \n\\begin{displaymath}\n\\bar{\\delta}=\\limsup_{i\\rightarrow \\infty}\\  {\\tt s}(p_{i})-r(p_{i})+2m.\n\\end{displaymath}\n\\end{Definition}\n\n\\vs \n\\noindent Note that coordinate-distance lags are always zero in the Schwarschild solution. From the next Proposition it will follow that coordinate-distance lags are always finite.\n\t\n\\begin{Proposition}\\label{CDLF} Let $(\\Sigma,{\\tt g},\\ln N)$ be an asymptotically flat static solution with connected and regular horizon $H$. Let $\\{\\bar{x}=(x_{1},x_{2},x_{3}),\\ |\\bar{x}|=r\\geq r_{1}\\}$ be a coordinate system as in Proposition \\ref{PBS2}. Then there are finite $c_{1}>c_{2}$, depending on $(\\Sigma,{\\tt g},\\ln N)$, with the following property: for every divergent sequence of points $\\{p_{i}\\}$ (lying inside the range of $\\{\\bar{x}\\}$) we have\n\\begin{equation}\\label{CDL}\n{\\tt s}(p_{i})-c_{2}\\leq r(p_{i})\\leq {\\tt s}(p_{i})-c_{1}.\n\\end{equation}\n\\end{Proposition} \n\n\\noindent {\\it Proof:}\n\n\\vs\n\tWe start showing the first inequality in equation (\\ref{CDL}). Let us first consider $r_{2}$ such that for every $\\bar{x}$ such that $r(|x|)\\geq r_{2}$ and a tangent vector $v$ at $\\bar{x}$ we have\n\\begin{displaymath}\n\\frac{|R(v,v)|}{|{\\tt g}_{S}(v,v)|}\\leq \\frac{R_{0}}{r^{3}}\\leq 1,\n\\end{displaymath}\n \n\\noindent where $R$ is the remainder tensor $R:={\\tt g}-{\\tt g}_{S}$, ${\\tt g}_{S}$ is the Schwarsdchild metric (\\ref{SchP}) and $R_{0}$ is a positive constant. It is clear that we do not loose anything in assuming that $r_{2}=r_{1}$.  \n\nLet $d_{0}=sup_{q\\in S_{r_{2}}}\\{dist(q,H)\\}$ and for each $i\\geq 0$ consider the curve $\\alpha(r)=(r,\\theta(p_{i})),\\varphi(p_{i}))$ starting at $S_{r_{2}}$ and ending at $p_{i}$ (namely the range of $r$ is $[r_{2},r(p_{i})]$. We will make use of the inequality \n\\begin{equation}\\label{SQE}\n\\sqrt{1+x}\\leq 1+|x|,\\ {\\rm if}\\ |x|<1,  \n\\end{equation}\n\n\\noindent to estimate the distance ${\\tt s}(p_{i})$ from above. We have\n\\begin{equation}\\label{SEA}\n{\\tt s}(p_{i})\\leq d_{0}+\\int_{r_{2}}^{r(p_{i})} \\sqrt{{\\tt g}_{S}(\\alpha',\\alpha')+R(\\alpha',\\alpha')}dr.\n\\end{equation}\n\n\\noindent As the integration is on $[r_{2},r(p_{i})]$ we have, by the definition of $r_{2}$, $|R(\\alpha',\\alpha')|\/|{\\tt g}_{S}(\\alpha',\\alpha')|\\leq R_{0}\/r^{3}\\leq 1$ (note that $\\alpha'=\\partial_{r}$). Thus by inequality (\\ref{SQE}) we have\n\\begin{displaymath} \n\\sqrt{{\\tt g}_{S}(\\alpha',\\alpha')+R(\\alpha',\\alpha')}\\leq \\sqrt{{\\tt g}(\\alpha',\\alpha')}+\\frac{R_{0}}{r^{3}}.\n\\end{displaymath}\n\n\\noindent Putting this into equation (\\ref{SEA}) and integrating we have\n\\begin{displaymath}\n{\\tt s}(p_{i})\\leq r(p_{i}) + (d_{0}+\\frac{R_{0}}{2r_{2}^{2}}-r_{2}).\n\\end{displaymath}\n\n\\noindent This proves the first inequality. \n\nTo show the second inequality on the right hand side of equation (\\ref{CDL}) we proceed as follows. Consider now an arbitrary curve $\\alpha(\\tau)$ joining $S_{r_{2}}$ to $p_{i}$, lying inside the region enclosed by $S_{r_{2}}$ and $S_{r(p_{i})}$ and parameterized by the arc length, with respect to ${\\tt g}_{S}$, $\\tau$. Then, for the length of $\\alpha$, $l(\\alpha)$, we have \n\\begin{displaymath}\nl(\\alpha)=\\int \\sqrt{{\\tt g}_{S}(\\alpha',\\alpha')+R(\\alpha',\\alpha')}d\\tau.\n\\end{displaymath}\n\n\\noindent We are going to make use of the inequality \n\\begin{equation}\\label{SQE2}\n1-|x|\\leq \\sqrt{1+x},\\ {\\rm if}\\ |x|\\leq 1.\n\\end{equation}\n\n\\noindent Note that because ${\\tt g}_{S}(\\alpha',\\alpha')=1$ we have $|R(\\alpha',\\alpha')|\\leq R_{0}\/r^{3}$. Therefore, from the inequality (\\ref{SQE2}) we have \n\\begin{equation}\\label{SI}\nl(\\alpha)\\geq \\int (1-\\frac{R_{0}}{r^{3}})d\\tau.\n\\end{equation}\n\n\\noindent Now note that $|dr\/d\\tau|\\leq 1$. To see this consider an arbitrary parameterization of $\\alpha$ by, say $t$. Then $d\\tau\/dt=\\sqrt{{\\tt g}_{S}(\\partial_{t}\\alpha,\\partial_{t}\\alpha)}\\geq |dr\/dt|$. \nThus, noting that the integrand in equation (\\ref{SI}) is positive, we can write\n\\begin{displaymath}\nl(\\alpha)\\geq \\int (1-\\frac{R_{0}}{r^{3}})d\\tau\\geq \\int (1-\\frac{R_{0}}{r^{3}})|\\frac{dr}{d\\tau}|d\\tau\\geq \\int (1-\\frac{R_{0}}{r^{3}})\\frac{dr}{d\\tau}d\\tau.\n\\end{displaymath}\n\t\n\\noindent Integrating we get\n\\begin{equation}\\label{FF}\nl(\\alpha)\\geq r_{i}-r_{2}-\\frac{R_{0}}{2r_{2}^{2}}.\n\\end{equation}\n\n\\noindent Now clearly we have ${\\tt s}(p_{i})$ is greater or equal than the infimum of the lengths of all the curves $\\alpha$ joining $p_{i}$ to $S_{r_{2}}$ and lying inside the region enclosed by $S_{r_{2}}$ and $S_{r(p_{i})}$. By the estimation in equation (\\ref{FF}) above we have thus \n\\begin{displaymath}\n{\\tt s}(p_{i})\\geq r(p_{i})-(r_{2}+\\frac{R_{0}}{2r_{2}}).\n\\end{displaymath}\n\n\\noindent which proves the inequality on the right hand side of equation (\\ref{CDL}).\\hspace{\\stretch{1}}$\\Box$ \t\n \t\n\\begin{Corollary}\nLet $(\\Sigma,{\\tt g},\\ln N)$ be an asymptotically flat static solution with connected and regular horizon $H$. Let $\\{\\bar{x}=(x_{1},x_{2},x_{3}),\\ |\\bar{x}|=r\\geq r_{1}\\}$ be a coordinate system as in Proposition \\ref{PBS2}. There are $c_{1}>c_{2}$ depending on $(\\Sigma,{\\tt g},\\ln N)$ with the following property: for every diverging sequence of points $\\{p_{i}\\}$ (lying inside the range of $\\{\\bar{x}\\}$) we have\n\\begin{displaymath}\nc_{2}\\leq \\bar{\\delta}(\\{p_{i}\\})\\leq c_{1}.\n\\end{displaymath}\n\\end{Corollary} \n\n\\subsection{Distance comparison.}\\label{SDC}\n\t\nConsider an asymptotically flat static solution with regular and connected horizon, $(\\Sigma,{\\tt g},\\ln N)$. Let ${\\tt s}(p)=dist(p,H)$. If the the solution $(\\Sigma,{\\tt g},\\ln N)$ were the Schwarzschild solution then we would have \n\\begin{displaymath}\n{\\tt s}(p)=r(p)-2m=\\frac{2m}{1-N(p)^{2}}-2m.\n\\end{displaymath}\n\n\\noindent As it turns out, given an arbitrary solution $(\\Sigma,{\\tt g},\\ln N)$, the function $\\hat{{\\tt s}}$ defined exactly by \n\\begin{displaymath}\n\\hat{{\\tt s}}(p):=\\frac{2m}{1-N(p)^{2}}-2m,\n\\end{displaymath}\n\n\\noindent provides, via a {\\it comparison of Laplacians}, a lower bound for the distance function ${\\tt s}$. The next Proposition computes the expression of the Laplacian of $\\hat{{\\tt s}}$. \n\\begin{Proposition} Let $(\\Sigma,{\\tt g},\\ln N)$ be a static solution of the Einstein equations. Then, the Laplacian of $\\hat{{\\tt s}}$ has the following expression\n\\begin{equation}\\label{SBF}\n\\Delta \\hat{{\\tt s}}=\\frac{2}{\\bar{s}+2m}(1+\\frac{m}{(\\hat{{\\tt s}}+2m)N^{2}})|\\nabla \\hat{{\\tt s}}|^{2}.\n\\end{equation}\n\\end{Proposition}\n\t\n\\vs\n\\noindent {\\it Proof:}\n\n\\vs\nNote first the identities \n\\begin{equation}\\label{SBI}\n\\hat{{\\tt s}} =2m\\frac{N^{2}}{1-N^{2}},\n\\end{equation}\n\\begin{equation}\\label{SBI2}\nN^{2}=\\frac{\\hat{{\\tt s}}}{\\hat{{\\tt s}}+2m},\\ N^{2}+1=2\\frac{\\hat{{\\tt s}}+m}{\\hat{{\\tt s}}+2m}. \n\\end{equation}\n\n\\noindent We calculate\n\\begin{displaymath}\n\\nabla \\frac{1}{1-N^{2}}=2\\frac{N\\nabla N}{(1-N^{2})^{2}}=2\\frac{N^{2}}{(1-N^{2})^{2}}\\nabla \\ln N.\n\\end{displaymath}\n\t\n\\noindent Next we compute the divergence of this expression to get\n\\begin{displaymath}\n\\Delta \\frac{1}{1-N^{2}}=4\\frac{|\\nabla N|^{2}}{(1-N^{2})^{2}} + 8 \\frac{N^{2}|\\nabla N|^{2}}{(1-N^{2})^{3}}=4\\frac{|\\nabla N|^{2}}{(1-N^{2})^{3}}(1+N^{2}),\n\\end{displaymath}\n\n\\noindent where we have used the fact that $\\Delta \\ln N=0$. This expression is equal to\n\\begin{displaymath}\n\\Delta \\frac{1}{1-N^{2}}=|\\nabla \\frac{1}{1-N^{2}}|^{2}(1+N^{2})(\\frac{1-N^{2}}{N^{2}}).\n\\end{displaymath}\n\n\\noindent After inserting back the coefficient $2m$ and using the identity (\\ref{SBI}) we get\n\\begin{displaymath}\n\\Delta \\hat{{\\tt s}}=\\frac{1+N^{2}}{\\hat{{\\tt s}}}|\\nabla \\hat{{\\tt s}}|^{2}.\n\\end{displaymath}\n\n\\noindent Finally, using the identity (\\ref{SBI2}) we have\n\\begin{displaymath}\n\\frac{N^{2}+1}{\\hat{{\\tt s}}}=2\\frac{\\hat{{\\tt s}}+m}{\\hat{{\\tt s}}+2m}\\frac{1}{\\hat{{\\tt s}}}=\\frac{2}{\\hat{{\\tt s}}+2m}(1+\\frac{m}{(\\hat{{\\tt s}}+2m)N^{2}}).\n\\end{displaymath}\n\\hspace{\\stretch{1}}$\\Box$\t\n\nThe asymptotic behavior of $\\hat{{\\tt s}}(p)$, when $r(p)\\rightarrow \\infty$ is deduced from Proposition \\ref{PBS2} and we have \n\\begin{equation}\\label{SBA}\n\\hat{{\\tt s}}(p)=\\frac{2m}{\\frac{2m}{r(p)}+O(\\frac{1}{r(p)^{3}})}-2m=r(p)-2m+O(\\frac{1}{r(p)}),\n\\end{equation}\n\n\\noindent if $r(p)$ is big enough. This asymptotic expression will be important and will be used many times later. \t\n\t\nThe reason why we have expressed the Laplacian of $\\hat{{\\tt s}}$ in the form (\\ref{SBF}) was to make it comparable with the Laplacian of ${\\tt s}$, that satisfies the inequality \n\\begin{equation}\\label{SL}\n\\Delta {\\tt s}\\leq \\frac{2}{{\\tt s}+2Pm}(1+\\frac{Pm}{({\\tt s} +2Pm)N^{2}})|\\nabla {\\tt s}|^{2}.\n\\end{equation}\n\n\\noindent in a certain {\\it barer sense} as is explained in Proposition \\ref{LSH}. In the equation above $P$ is equal to the expression\n\\begin{displaymath}\nP=\\frac{A}{16\\pi m^{2}},\n\\end{displaymath}\n\n\\noindent and will be called {\\it the Penrose quotient}. Note that the Penrose inequality $A\\leq 16\\pi m^{2}$ holds iff $P\\leq 1$. Note too that wherever $s$ is smooth we have $|\\nabla s|^{2}=1$. We have included such factor in (\\ref{SL}) to make the comparison to (\\ref{SBF}) more evident. \n\n\nThe fact that the inequality (\\ref{SL}) holds in a barer sense will allow us to assume, when comparing ${\\tt s}$ to $\\hat{{\\tt s}}$, that ${\\tt s}$ is a smooth function. This fact will be further explained in Theorem \\ref{DC}. We now introduce a Proposition describing the {\\it sense} in which inequality (\\ref{SL}) holds.\n\n\\begin{Proposition}\\label{LSH}\nLet $(\\Sigma,{\\tt g},\\ln N)$ be an asymptotically flat static solution with regular and connected horizon. Let $\\{p_{i}\\}_{i=1}^{i=\\infty}$ be a sequence of points in $\\Sigma$ converging to $p$ in $\\Sigma\\setminus H$. Let $\\{\\Gamma_{i}\\}_{i=1}^{i=\\infty}$ be a sequence such that $\\lim_{i\\rightarrow 0} \\Gamma_{i}\\downarrow 0$, $\\Gamma_{1}\\leq \\Gamma_{0}$ with $\\Gamma_{0}$ regular (Notation \\ref{NOT1}) and $\\{p_{i},i=1,\\ldots,i=\\infty\\}\\subset \\Sigma\\setminus \\Omega_{H,H_{\\Gamma_{0}}}$. Consider the sequence of distance functions $\\{{\\tt s}_{\\Gamma_{i}}(p)=dist(p,H_{\\Gamma_{i}})\\}_{i=1}^{i=\\infty}$. Then, there is sequence of continuous functions $\\tilde{{\\tt s}}_{\\Gamma_{i}}$ such that for each $\\Gamma_{i}$:\n\\begin{enumerate}\n\\item $\\tilde{{\\tt s}}_{\\Gamma_{i}}$ is defined on the domain $\\Sigma\\setminus \\Omega_{H,H_{\\Gamma_{i}}}$,\n\n\\item $\\tilde{{\\tt s}}_{\\Gamma_{i}}$ is smooth at $p_{i}$, \n\n\\item $\\tilde{{\\tt s}}_{\\Gamma_{i}}\\geq {\\tt s}_{\\Gamma_{i}}$, $\\tilde{{\\tt s}}_{\\Gamma_{i}}(p_{i})= {\\tt s}_{\\Gamma_{i}}(p_{i})$ and $|\\nabla \\tilde{{\\tt s}}|^{2}(p_{i})=1$.\n\n\\item\n\\begin{displaymath}\n\\Delta \\tilde{{\\tt s}}_{\\Gamma_{i}}(p_{i})\\leq 2\\frac{1}{\\tilde{{\\tt s}}_{\\Gamma_{i}}(p_{i})+\\tilde{a}_{i}}(1+\\frac{\\tilde{a}_{i}}{2(\\tilde{{\\tt s}}_{\\Gamma_{i}}(p_{i})+\\tilde{a}_{i}) N^{2}(p_{i})})|\\nabla \\tilde{{\\tt s}}_{\\Gamma_{i}}|^{2}(p_{i}),\n\\end{displaymath}\n\\noindent where $\\{\\tilde{a}_{i}\\}$ is a sequence such that $\\lim_{i\\rightarrow \\infty} \\tilde{a}_{i}=2mP$.\n\\item  Moreover, $\\{\\tilde{{\\tt s}}_{\\Gamma_{i}}\\}$ converges uniformly in $C^{0}$ to ${\\tt s}(p)=dist(p,H)$ in the sense that \n\\begin{displaymath}\n\\lim_{i\\rightarrow \\infty} \\sup_{q\\in \\Sigma\\setminus \\Omega_{H,H_{\\Gamma_{i}}}}|\\tilde{{\\tt s}}_{\\Gamma_{i}}(q)-{\\tt s}(q)|=0.\n\\end{displaymath}\n\\end{enumerate}\n\\end{Proposition}\nThe proof of this Proposition will be a direct consequence of the following Proposition in Riemannian geometry. We will use the following notation and terminology. \n\\begin{Notation}\\label{NOT2}\nLet $(\\Sigma, g)$ be a complete Riemannian manifold with non-empty and connected boundary $\\partial \\Sigma$. The {\\it inner-normal bundle} ${\\mathcal{N}}(\\partial \\Sigma)$ of $\\Sigma$ at $\\partial \\Sigma$ is defined as the set of vectors $v(q)$, normal to $\\partial \\Sigma$ at $q$, and pointing inwards to $\\Sigma$. We will consider the exponential map $exp:{\\mathcal{N}}(\\partial \\Sigma)\\rightarrow \\Sigma$ such that to every $v(q)\\in {\\mathcal{N}}(\\partial \\Sigma)$ assigns the end point of the geodesic segment of length $|v(q)|$ that start at $q$ with velocity $v(q)\/|v(q)|$. \n\\end{Notation}\n\n\\begin{Proposition}\\label{LLSH}\nLet $(\\Sigma, g)$ be a complete Riemannian three-manifold, not necessarily compact. Let ${\\mathcal{S}_{1}}$ be an immersed smooth surface separating $\\Sigma$ into two connected (open) components $\\Sigma_{1}$ and $\\Sigma_{2}$. Let $p$ be a point in $\\Sigma_{1}$ and $\\gamma_{q,p}$ be a geodesic segment minimizing the distance between $p$ and $\\partial \\Sigma_{1}={\\mathcal{S}_{1}}$, starting at $q\\in \\partial \\Sigma_{1}$ and ending at $p$. We can write $\\gamma_{q,p}(\\tau)=exp(\\tau v(q))$, $\\tau\\in[0,1]$, with $v(q)=l(\\gamma_{q,p})n(q)$  where $n(q)$ is the inward unit-normal vector to $\\partial \\Sigma_{1}$ at $q$. If the differential of the exponential map $exp:{\\mathcal{N}}(\\partial \\Sigma_{1})\\rightarrow \\Sigma_{1}$ is not injective at $v(q)$, then for every smooth surface ${\\mathcal{S}}_{2}$ immersed in $\\Sigma_{1}\\cup {\\mathcal{S}}_{1}$ such that \n\\begin{enumerate}\n\\item ${\\mathcal{S}}_{2}$ touches ${\\mathcal{S}}_{1}$ only at $q$,\n\n\\item The second fundamental forms $\\Theta_{1}(q)$ and $\\Theta_{2}(q)$ of ${\\mathcal{S}}_{1}$ and ${\\mathcal{S}}_{2}$ (respectively) at $q$ and defined with respect to $n(q)$ satisfy\n\\begin{displaymath}\n\\Theta_{2}(q)>\\Theta_{1}(q).\n\\end{displaymath}\n\\end{enumerate}\n\n\\noindent we have, \n\\begin{enumerate}\n\\item $\\gamma_{q,p}$ is the only geodesic segment minimizing the distance between $p$ and ${\\mathcal{S}}_{2}$,\n\n\\item The exponential map $exp:{\\mathcal{N}}(\\partial \\tilde{\\Sigma}_{1})\\rightarrow \\tilde{\\Sigma}_{1}$ is injective at $v(q)$, where $\\tilde{\\Sigma}_{1}$ is the connected component of $\\Sigma\\setminus {\\mathcal{S}}_{2}$ containing ${\\mathcal{S}}_{1}$.\n\\end{enumerate}\n\\end{Proposition}\n\n\\noindent {\\it Proof:}\n\n\\vs\nFirst it is clear that $\\gamma_{p,q}$ is the only geodesic segment minimizing the distance between $p$ and ${\\mathcal{S}}_{2}$ for ${\\mathcal{S}}_{2}$ touches ${\\mathcal{S}}_{1}$ only at $q$. This proves the first {\\it item} of the claim. \n\nTo prove the second suppose on the contrary that the exponential map $exp:{\\mathcal{N}}(\\partial \\tilde{\\Sigma}_{1})\\rightarrow \\tilde{\\Sigma}_{1}$ is not injective at $v(q)$. Then there is a curve $w(\\lambda)$, $\\lambda\\in [0,\\lambda_{1}]$ of vectors in ${\\mathcal{N}}(\\tilde{\\Sigma})$ of norm (for all $\\lambda$) equal to $l(\\gamma_{p,q})$, such that $w(0)=v(q)$ and such that $d\\ exp (w'(0))=0$. Therefore $J(s)=d\\ exp(\\frac{s}{l(\\gamma_{p,q})}w'(0)$ is a Jacobi field such that $J(s)\\neq 0$ for any $s\\in [0,l(\\gamma_{p,q}))$. Let $\\alpha(s,\\lambda)$, $(s,\\lambda)\\in [0,l(\\gamma_{p,q})]\\times [0,\\lambda_{1}]$ be a smooth one-parameter family of curves such that $\\partial_{\\lambda} \\alpha (s,0)=J(s)$ and such that $\\partial_{s}\\alpha (0,\\lambda)\\in {\\mathcal{N}}(\\tilde{\\Sigma}_{1})$. Then because $J(s)$ is a Jacobi field we have that the second variation of the length of the curves $\\alpha_{\\lambda}(s)=\\alpha(s,\\lambda)$ (variation with respect to $\\lambda$) is equal to zero\\footnote{Although it is a standard fact in Riemannian geometry, the reader can check this fact in pages 227-228 of \\cite{MR757180}. The proof there is for Jacobi fields vanishing at the two extreme points, but it is simply adapted to this situation as well.}. On the other hand consider the curves $\\bar{\\alpha}(s,\\lambda)=\\alpha(s,\\lambda)$, with $(s,\\lambda)\\in [0,s(\\lambda)]\\times [0,l(\\gamma_{p,q})]$ where the point $\\alpha(s(\\lambda),\\lambda)$ is the intersection of $\\alpha(s,\\lambda)$ (a curve as a function of $s$) and ${\\mathcal{S}}_{1}$. Now, because of the conditon in {\\it item 2}, $\\Theta_{2}(q)>\\Theta_{1}(q)$, the second variation (with respect to $\\lambda$) of $\\bar{\\alpha}$ is positive. Thus the second variation (with respect to $\\lambda$) of the length of the curves $\\tilde{\\alpha}(s,\\lambda)=\\alpha(s,\\lambda)$, $(s,\\lambda)\\in [s(\\lambda),l(\\gamma_{p,q})]\\times [0,\\lambda_{1}]$ is negative, which is a contradiction as $\\gamma_{p,q}$ is length minimizing between $p$ and ${\\mathcal{S}}_{1}$.\\hspace{\\stretch{1}}$\\Box$      \n\n\\vs\n\\noindent {\\it Proof ({\\rm{\\bf of Proposition \\ref{LSH}}})}:\n\n\\vs\nLet $\\gamma_{p_{i},q_{i}}$, $q_{i}\\in H_{\\Gamma_{i}}$ be a length minimizing geodesic joining $p_{i}$ and $H_{\\Gamma_{i}}$. Suppose first that ${\\tt s}_{\\Gamma_{i}}$ is smooth at $p_{i}$ for each $i$. Then we claim that taking $\\tilde{{\\tt s}}_{\\Gamma_{i}}={\\tt s}_{\\Gamma_{i}}$ is enough. It is clear that the {\\it items 1,2,3} and {\\it 5} of the claim are satisfied with this choice.  We need therefore to check that there is sequence $\\tilde{a}_{i}$ for which the equation in {\\it item 4} is satisfied and $\\lim_{i\\rightarrow \\infty} \\tilde{a}_{i}=2mP$. For this we are going to use the monotonicity, for every $a$ of ${\\mathcal{M}}={\\mathcal{M}}_{a}$ an over $\\gamma_{p_{i},q_{i}}$, and then we will chose $a$ conveniently (which will be our choice of $\\tilde{a}_{i}$). Of course ${\\mathcal{M}}$ is defined, for each $i$, for the congruences ${\\mathcal{F}}_{i}$ of length minimizing geodesics segments to $H_{\\Gamma_{i}}$. Thus we have\n\\begin{displaymath}\n\\frac{\\theta(p_{i})}{2}({\\tt s}_{\\Gamma_{i}}(p_{i})+a)^{2}N^{2}(p_{i})-({\\tt s}_{\\Gamma_{i}}(p_{i})+a)N^{2}(p_{i})={\\mathcal{M}}_{a}(p_{i})\\leq {\\mathcal{M}}_{a}(q_{i}).\n\\end{displaymath} \n\n\\noindent Solving for $\\theta(p_{i})=\\Delta {\\tt s}_{\\Gamma_{i}} (p_{i})$ we get\n\\begin{displaymath}\n\\Delta {\\tt s}_{\\Gamma_{i}}(p_{i})\\leq \\frac{2}{({\\tt s}_{\\Gamma_{i}}(p_{i})+a)}(1+\\frac{{\\mathcal{M}}_{\\Gamma_{i}}(q_{i})}{({\\tt s}_{\\Gamma_{i}}(p_{i})+a)N^{2}(p_{i})}).\n\\end{displaymath}\n\n\\noindent We need now to show that we can chose $a$ for each $i$ (thus having $a=\\tilde{a}_{i}$) in such a way that    \n${\\mathcal{M}}_{\\Gamma_{i}}(q_{i})\\leq \\tilde{a}_{i}\/2$. Therefore we need to have\n\\begin{displaymath}\n{\\mathcal{M}}_{\\Gamma_{i}}(q_{i})=\\frac{\\theta_{\\Gamma_{i}}(q_{i})}{2}a^{2}N^{2}(q_{i})-aN(q_{i})^{2}\\leq \\frac{a}{2}.\n\\end{displaymath}\n\n\\noindent Thus we chose\n\\begin{equation}\\label{ADEF}\na= \\sup_{q\\in H_{\\Gamma_{i}}} \\{\\frac{2(\\frac{1}{2}+N(q)^{2})}{\\theta(q)N^{2}(q)}\\}.\n\\end{equation}\n\n\\noindent Now, the numerator tends to one and the denominator, because of equation (\\ref{MH}), tends to $2|\\nabla N|_{H}=8\\pi m\/A=1\/(2mP)$. The claim in this case follows. \n\nIf on the contrary the functions ${\\tt s}_{\\Gamma_{i}}$ are not smooth at $p_{i}$, then we know by Proposition \\ref{LLSH} that the distance functions $\\tilde{{\\tt s}}_{\\Gamma_{i}}$ to a hypersurface $\\tilde{H}_{\\Gamma_{i}}$ included in $\\Omega_{H,H_{\\Gamma_{i}}}$ will be smooth at $p_{i}$ provided they touch $H_{\\Gamma_{i}}$ only at $q_{i}$ and have strictly grater second fundamental form at $q_{i}$. Besides these last two conditions nothing else is required on the hypersurfaces $\\tilde{H}_{\\Gamma_{i}}$ for $\\tilde{{\\tt s}}_{\\Gamma_{i}}$ to be smooth at $p_{i}$. Thus, it is clear that if we chose the hypersurfaces $\\tilde{H}_{\\Gamma_{i}}$ close enough to $H_{\\Gamma_{i}}$ (but satisfying the two requirements) and $\\tilde{a}_{i}$ using the same formula as in equation \\ref{ADEF} (but with $q$ varying on $\\tilde{H}_{\\Gamma_{i}}$) then $\\tilde{{\\tt s}}_{\\Gamma_{i}}$ will satisfy {\\it items 1- 5} of the claim. \\hspace{\\stretch{1}}$\\Box$\n\n\\begin{Theorem}\\label{DC}{\\rm{\\bf (Distance comparison).}} Let $(\\Sigma,{\\tt g},\\ln N)$ be an asymptotically flat static solution with regular and connected horizon. Then we have \n\\begin{equation}\\label{MDC}\n\\frac{2m}{1-N^{2}(p)}-2m=\\hat{{\\tt s}}(p)\\leq \\max\\{1,\\frac{1}{P}\\}{\\tt s}(p)=\\max\\{1,\\frac{16 \\pi m^{2}}{A}\\} dist(p,H),\n\\end{equation}\n\n\\noindent for all $p$ in $\\Sigma$, where $P$ is the Penrose quotient. Moreover \n\\begin{displaymath}\n\\lim_{{\\tt s}(p)\\rightarrow \\infty} \\frac{\\hat{{\\tt s}}(p)}{{\\tt s}(p)}=1,\\ {\\rm and}\\ \\lim_{{\\tt s}(p)\\rightarrow 0} \\frac{\\hat{{\\tt s}}(p)}{{\\tt s}(p)}=\\frac{1}{P},\n\\end{displaymath}\n\\end{Theorem}    \n\n\\noindent {\\it Proof:}\n\n\\vs\nWe will consider the quotient $\\hat{{\\tt s}}\/{\\tt s}$ as a function on $\\Sigma\\setminus H$. Let us first find the boundary conditions, namely $\\lim \\hat{{\\tt s}}(p)\/{\\tt s}(p)$ when ${\\tt s}(p)\\rightarrow \\infty$ and ${\\tt s}(p)\\rightarrow 0$ (at infinity and at the horizon respectively). From Proposition \\ref{CDLF} and the estimation (\\ref{SBA}) we deduce \n\\begin{displaymath}\n\\lim_{{\\tt s}(p)\\rightarrow \\infty} \\frac{\\hat{{\\tt s}}(p)}{{\\tt s}(p)}=1. \n\\end{displaymath}  \n\t\n\\noindent To calculate the quotient at the horizon we proceed like this. Consider the congruence of geodesics with respect to $g$, emanating perpendicularly to $H$ and parameterized by the arc length $\\tau$ which is measured from the initial point of the geodesic at $H$. Any given coordinate system $\\{\\bar{x}=(x_{1},x_{2})\\}$ on an open set of $H$ can be propagated along the congruence to the level sets of the distance function with respect to $g$, namely the $\\tau_{0}$-level sets $\\{\\tau=\\tau_{0}\\}$ and we can write\n\\begin{displaymath}\ng=d\\tau^{2}+h_{ij}(\\bar{x},\\tau)dx_{i}dx_{j},\n\\end{displaymath}\n\n\\noindent and \n\\begin{equation}\\label{CCC}\n\\hat{{\\tt s}}(\\tau,\\bar{x})=\\frac{2m}{1-N^{2}(\\tau,\\bar{x})}-2m=2m|\\nabla N|^{2}_{H}\\tau^{2}+O(\\tau^{3}).\n\\end{equation}\n\n\\noindent We note then that because $H$ is totally geodesic, the second fundamental form is zero and we have \n\\begin{displaymath}\n\\partial_{\\tau} h_{ij}(\\tau,\\bar{x})\\bigg|_{\\tau=0}=0.\n\\end{displaymath}\n\n\\noindent Thus\n\\begin{equation}\\label{CCCC}\ng=d\\tau^{2}+h_{ij}(0,\\bar{x})dx_{i}dx_{j}+O(\\tau^{2}).\n\\end{equation}  \n\n\\noindent Combining (\\ref{CCC}) and (\\ref{CCCC}) we get \n\\begin{displaymath}\n{\\tt g}=N^{2}g=|\\nabla N|_{H}^{2}\\tau^{2}(d\\tau^{2}+h_{ij}(0,\\bar{x})dx_{i}dx_{j}) + O(\\tau^{3})d\\tau^{2}+O(\\tau^{4})h_{ij}dx_{i}dx_{j}.\n\\end{displaymath}\n\n\\noindent From this expression it is simple that if $\\{p_{i}\\}$ is a sequence in $\\Sigma\\setminus H$ converging to a point in $H$ we have\n\\begin{equation}\\label{C4}\n{\\tt s}(p_{i})=|\\nabla N|_{H}\\frac{\\tau(p_{i})^{2}}{2}+O(\\tau(p_{i})^{3}).\n\\end{equation}\n\n\\noindent We can combine (\\ref{CCC}) and (\\ref{C4}) to conclude that for any sequence $\\{p_{i}\\}$ in $\\Sigma\\setminus H$ converging to a point in $H$ we have\n\\begin{equation}\\label{C5}\n\\lim \\frac{\\hat{{\\tt s}}(p_{i})}{{\\tt s}(p_{i})}=4m|\\nabla N|_{H}.\n\\end{equation}\n\n\\noindent Now, $|\\nabla N|_{H}$ is equal to $4\\pi m\/A$ as can be seen by integrating $\\Delta N=0$ between $S_{r}=\\{p\/r(p)=r\\}$ and $H$ and taking the limit when $r\\rightarrow \\infty$. With this value of $|\\nabla N|_{H}$ we get from (\\ref{C5})\n\\begin{displaymath}\n\\lim_{{\\tt s}(p)\\rightarrow 0} \\frac{\\hat{{\\tt s}}(p)}{{\\tt s}(p)}=\\frac{16\\pi m^{2}}{A}=\\frac{1}{P}.\n\\end{displaymath}   \n\nWe would like now to compare $\\hat{{\\tt s}}$ to ${\\tt s}$ using (\\ref{SBF}) and (\\ref{SL}). For this purpose it is simpler to consider the dimensionless quantities $\\hat{u}=\\hat{{\\tt s}}\/2m$ and $u={\\tt s}\/2mP$. In terms of them (\\ref{SBF}) and (\\ref{SL}) become\n\\begin{equation}\\label{BSIU0}\n\\Delta \\hat{u}=\\frac{2}{\\hat{u}+1}(1+\\frac{1}{2(\\hat{u}+1)N^{2}})|\\nabla \\hat{u}|^{2},\n\\end{equation}\n\\begin{equation}\\label{BSIU}\n\\Delta u\\leq\\frac{2}{u+1}(1+\\frac{1}{2(u+1)N^{2}})|\\nabla u|^{2},\n\\end{equation}\n\n\\noindent We will consider now the quotient $\\phi=\\hat{u}\/u$ and note that the boundary conditions at $H$ and at infinity become, respectively, $\\lim_{{\\tt s}(p)\\rightarrow 0}\\hat{u}(p)\/u(p)=1$ and $\\lim_{{\\tt s}(p)\\rightarrow \\infty}\\hat{u}(p)\/u(p)=P$. If we prove that $\\hat{u}\/u\\leq \\max\\{1,P\\}$ then we will be proving (\\ref{MDC}). Thus we will proceed by contradiction and assume that there is a point $\\bar{p}\\in \\Sigma\\setminus H$ such that $\\hat{u}(\\bar{p})>\\max\\{1,P\\}u(\\bar{p})$ and that such point is an absolute maximum for $\\hat{u}\/u$ (note the boundary conditions). We will assume below that the function ${\\tt s}$ is smooth at $\\bar{p}$, or, equivalently that $u$ is smooth at $\\bar{p}$. Otherwise use the fact that ${\\tt s}$ satisfies equation (\\ref{SL}) in a barer sense as follows. Replace ${\\tt s}$ by ${\\tt s}_{\\Gamma}$ for $\\Gamma$ sufficiently small in such a way that $\\hat{u}\/u_{\\Gamma}$, with $u_{\\Gamma}={\\tt s}_{\\Gamma}\/2mP$ still has a maximum greater than $\\max\\{1,P\\}$, say at $\\bar{\\bar{p}}$. Then substitute once more ${\\tt s}_{\\Gamma}$ by $\\tilde{{\\tt s}}_{\\Gamma}\\geq {\\tt s}_{\\Gamma}$ as in Proposition \\ref{LSH} and consider thus the quotient $\\hat{u}\/\\tilde{u}_{\\Gamma}$, with $\\tilde{u}_{\\Gamma}=\\tilde{{\\tt s}}_{\\Gamma}\/2mP$, which still has a maximum greater than $\\max\\{1,P\\}$ at $\\bar{\\bar{p}}$. If $\\Gamma$ is sufficiently small we would reach a contradiction following the same argument as below. \n\nWe compute\n\\begin{equation}\\label{CLQ}\n\\Delta \\frac{\\hat{u}}{u}=\\frac{\\Delta \\hat{u}}{u}-2\\frac{<\\nabla \\hat{u},\\nabla u>}{u^{2}}-\\frac{\\hat{u}}{u^{2}}\\Delta u+2\\frac{\\hat{u}}{u^{3}}|\\nabla u|^{2}.\n\\end{equation}\n   \n\\noindent Because $\\hat{u}\/{u}$ reaches an absolute maximum at $\\bar{p}$ we have $\\nabla (\\hat{u}\/u|_{\\bar{p}})=0$ and thus\n\\begin{equation}\\label{GEM}\n\\frac{\\nabla \\hat{u}}{\\hat{u}}\\bigg|_{\\bar{p}}=\\frac{\\nabla u}{u}\\bigg|_{\\bar{p}},\n\\end{equation}\n\n\\noindent with $|\\nabla u|^{2}(\\bar{p})=1\/2mP\\neq 0$. If we use (\\ref{GEM}) in (\\ref{CLQ}) we note that the second and fourth terms on the right hand side cancel out at $\\bar{p}$. Thus we will get a contradiction of the fact that $\\hat{u}\/u$ reaches an absolute maximum at $\\bar{p}$ if we can prove that the sum of the first and third terms on the right hand side of (\\ref{CLQ}) is positive at $\\bar{p}$ (the Maximum Principle). We will prove that in what follows.\n\nWe compute\n\\begin{displaymath}\n\\Delta \\frac{\\hat{u}}{u}\\ \\bigg|_{\\bar{p}}=\\frac{1}{u^{2}(\\bar{p})}(u\\Delta \\hat{u} -\\hat{u}\\Delta u)\\bigg|_{\\bar{p}}.\n\\end{displaymath}\n\n\\noindent and using (\\ref{BSIU0}) and (\\ref{BSIU}) we get the inequality\n\\begin{displaymath}\n\\Delta\\frac{\\hat{u}}{u}\\bigg|_{\\bar{p}}\\geq \\frac{2}{u^{2}}(\\frac{u}{(1+\\hat{u})}(1+\\frac{1}{2(1+\\hat{u})N^{2}}))\\frac{\\hat{u}^{2}}{u^{2}}|\\nabla u|^{2} - \\frac{\\hat{u}}{(1+u)}(1+\\frac{1}{2(1+u)N^{2}})|\\nabla u|^{2})\\bigg|_{\\bar{p}}.\n\\end{displaymath}\n\n\\noindent Thus we would like to prove that \n\\begin{equation}\\label{FOR}\n\\frac{\\hat{u}}{1+\\hat{u}}(1+\\frac{1}{2(1+\\hat{u})N^{2}})\\bigg|_{\\bar{p}}>\\frac{u}{1+u}(1+\\frac{1}{2(1+u)N^{2}})\\bigg|_{\\bar{p}}.\n\\end{equation}\n\t\n\\noindent Recalling from (\\ref{SBI2}) that $N^{2}=\\hat{u}\/(1+\\hat{u})$ and substituting that into (\\ref{FOR}) we deduce that we would like to show that\n\\begin{displaymath}\n\\frac{\\hat{u}}{(1+\\hat{u})}(1+\\frac{1}{2\\hat{u}})\\bigg|_{\\bar{p}}>\\frac{u}{1+u}(1+\\frac{1+\\hat{u}}{2(1+u)\\hat{u}})\\bigg|_{\\bar{p}}.\n\\end{displaymath} \n\t\t\n\\noindent We will arrange now this equation in a different form. To this, right hand term $u\/(1+u)$ is moved to the left hand side, while the left hand term $1\/(2(1+\\hat{u}))$ is moved to the right hand side. In this way we obtain a new inequality where the left hand side is\n\\begin{displaymath}\n\\frac{\\hat{u}}{1+\\hat{u}}-\\frac{u}{1+u}\\bigg|_{\\bar{p}}=\\frac{\\hat{u}-u}{(1+u)(1+\\hat{u})}\\bigg|_{\\bar{p}},\n\\end{displaymath}\n\n\\noindent and where the right hand side is \n\\begin{displaymath}\n\\frac{u(1+\\hat{u})}{2\\hat{u}(1+u)^{2}}-\\frac{1}{2(1+\\hat{u})}\\bigg|_{\\bar{p}}=\\frac{1}{2\\hat{u}(1+\\hat{u})(1+u)^{2}}(u(1+\\hat{u})^{2}-\\hat{u}(1+u)^{2})\\bigg|_{\\bar{p}}.\n\\end{displaymath}\n\n\\noindent This last expression can be further arranged into\n\\begin{displaymath}\n\\frac{1}{2\\hat{u}(1+\\hat{u})(1+u)^{2}}(\\hat{u}-u)(\\hat{u}u-1)\\bigg|_{\\bar{p}}.\n\\end{displaymath}\n\n\\noindent Thus combining the results on the left and right hands we conclude that we would like the inequality\n\\begin{displaymath}\n\\frac{\\hat{u}-u}{(1+u)(1+\\hat{u})}\\bigg|_{\\bar{p}}>\\frac{1}{2\\hat{u}(1+\\hat{u})(1+u)^{2}}(\\hat{u}-u)(\\hat{u}u-1)\\bigg|_{\\bar{p}},\n\\end{displaymath}\n\n\\noindent to be satisfied. Thus we would like to have\n\\begin{displaymath}\n2(\\hat{u}-u)\\hat{u}(1+u)\\bigg|_{\\bar{p}}> (\\hat{u}-u)(\\hat{u}u-1)\\bigg|_{\\bar{p}},\n\\end{displaymath}\n\n\\noindent but because we are assuming $\\hat{u}(\\bar{p})>\\max\\{1,P\\}u(\\bar{p})\\geq u(\\bar{p})$ the inequality above is clearly satisfied.\\hspace{\\stretch{1}}$\\Box$ \n\n\\subsection{The Penrose inequality.}\\label{SPI}\n\nIn this section we will prove the Penrose inequality for asymptotically flat static solutions with regular and connected horizon. We start by observing and interesting Corollary to Theorem \\ref{DC}.\n\n\\begin{Corollary} {\\rm (To Theorem \\ref{DC})}\\label{CDC}\nLet $(\\Sigma,{\\tt g},\\ln N)$ be an asymptotically flat static solution with connected and regular horizon. Suppose that the Penrose inequality does not hold, namely, assume that the Penrose quotient $P=\\frac{A}{16\\pi m^{2}}$ is greater than one. Then, for any divergent sequence of points $\\{p_{i}\\}$, the associated coordinate-distance lag is greater or equal than zero, namely $\\bar{\\delta}(\\{p_{i}\\})\\geq 0$.\n\\end{Corollary}\n\n\\noindent {\\it Proof:}\n\n\\vs\nIf $P>1$ then $\\max\\{1,\\frac{1}{P}\\}=1$ and from Theorem \\ref{DC} we have then \n\\begin{displaymath}\n\\hat{{\\tt s}}(p)=\\frac{2m}{1-N^{2}(p)}-2m\\leq {\\tt s}(p),\\ {\\rm for\\ all}\\ p\\in \\Sigma.\n\\end{displaymath}\n\n\\noindent Evaluating this inequality at $\\{p_{i}\\}$ and using the asymptotic of $\\hat{{\\tt s}}$ described in equation (\\ref{SBA}) we get\n\\begin{displaymath}\n 0\\leq {\\tt s}(p_{i})-r(p_{i})+2m+O(\\frac{1}{r(p_{i})}),\n\\end{displaymath}\n \n\\noindent Therefore\n\\begin{displaymath}\n0\\leq \\limsup_{i\\rightarrow \\infty} {\\tt s}(p_{i})-r(p_{i})+2m=\\bar{\\delta}(\\{p_{i}\\}).\n\\end{displaymath}\n\n\\noindent as desired.\\hspace{\\stretch{1}}$\\Box$\n\nThe following Proposition however shows (in particular) that if the Penrose inequality does not hold then there is a divergent sequence $\\{p_{i}\\}$ whose coordinate-distance lag is negative, namely $\\bar{\\delta}(\\{p_{i}\\})<0$. The two results thus show the Penrose inequality on asymptotically flat static solutions with regular and connected horizon.\n\n\\begin{Proposition}\\label{PDC}\nLet $(\\Sigma,{\\tt g},\\ln N)$ be an asymptotically flat static solution with regular and connected horizon $H$. Then, there is a divergent sequence $\\{p_{i}\\}$ such that \n\\begin{displaymath}\n\\bar{\\delta}(\\{p_{i}\\})\\leq m(1-P).\n\\end{displaymath}\n\n\\noindent In particular if $P>1$ then $\\bar{\\delta}(\\{p_{i}\\})<0$.\n\\end{Proposition}\n\n\\noindent {\\it Proof:} \n\n\\vs\nLet $\\{\\Gamma_{i}\\}_{i=1}^{i=\\infty}$ be a sequence such that $\\Gamma_{i}\\downarrow 0$ (with $\\Gamma_{1}\\leq \\Gamma_{0}$ and $\\Gamma_{0}$ regular as in Notation \\ref{NOT1}), and let $\\{r_{i}\\}_{i=1}^{i=\\infty}$ be a sequence such that $r_{i}\\uparrow \\infty$ (and $r_{1}$ as in Proposition \\ref{PBS2}). \nConsider the congruence of length minimizing geodesics ${\\mathcal{F}}$ emanating perpendicularly to $H_{\\Gamma_{i}}$. The geodesic segment, $\\gamma_{i}$, minimizing the length between $H_{\\Gamma_{i}}$ and $S_{r_{i}}$ is clearly in ${\\mathcal{F}}$. Let $p_{i}$ be the point of $\\gamma_{i}$ at $S_{r_{i}}$, let $q_{i}$ be the initial point at $H_{\\Gamma_{i}}$ and let $v(q_{i})$ the (unit) velocity of $\\gamma_{i}$ at $q_{i}$. $\\gamma_{i}$ is naturally perpendicular to $S_{r_{i}}$ at $p_{i}$ and to $H_{\\Gamma_{i}}$ at $q_{i}$. Consider now the exponential map $exp:{\\mathcal{N}}_{i}\\rightarrow \\Sigma$, where ${\\mathcal{N}}_{i}$ is the inner-normal bundle of $\\Sigma\\setminus \\Omega_{H,H_{\\Gamma_{i}}}$ at $H_{\\Gamma_{i}}$ as in Notation \\ref{NOT2}. Assume that the differential of the exponential map is smooth at the point $l(\\gamma_{i})v(q_{i})$ in ${\\mathcal{N}}_{i}$, if not, work instead with a suitable function $\\tilde{{\\tt s}}_{\\Gamma_{i}}$ as in Proposition \\ref{LSH}. Note that, in the notation of Proposition \\ref{LSH}, we have $l(\\gamma_{i})={\\tt s}_{\\Gamma_{i}}(p_{i})$. Then, there is $\\epsilon_{i}$ such that the surface defined by $\\bar{S}_{i}=\\{exp(l(\\gamma_{i})v(q)),\\ q\\in B_{H_{\\Gamma_{i}}}(q_{i},\\epsilon_{i})\\}$ is smooth. Moreover $\\bar{S}_{i}$ is tangent to $S_{r_{i}}$ at $p_{i}$, its mean curvature is equal to the mean curvature  $\\theta$ of ${\\mathcal{F}}$ restricted to it, and, because $\\gamma_{i}$ is length minimizing between $S_{r_{i}}$ and $H_{\\Gamma_{i}}$, it lies inside the region enclosed by $H_{\\Gamma_{i}}$ and $S_{r_{i}}$. Therefore from the standard comparison of mean curvatures we have\n\\begin{displaymath}\n\\theta(p_{i})\\geq \\theta_{r_{i}}(p_{i}),\n\\end{displaymath}\n\n\\noindent where $\\theta_{r_{i}}$ is the mean curvature of $S_{r_{i}}$. Consider now ${\\mathcal{M}}$ with $a=A\/8\\pi m$ and over $\\gamma_{i}$. As ${\\mathcal{M}}$ is monotonic we have\n\\begin{displaymath}\n\\theta(p_{i})\\leq \\frac{2}{{\\tt s}_{\\Gamma_{i}}+\\frac{A}{8\\pi m}} + \\frac{2{\\mathcal{M}}(q_{i})}{({\\tt s}_{\\Gamma_{i}}+\\frac{A}{8\\pi m})^{2}N^{2}(p_{i})}.\n\\end{displaymath}     \n\n\\noindent Now, to use this equation we need several facts. First, from Proposition \\ref{PMC} we have $\\theta_{r_{i}}=2\/r_{i}+2m\/(r_{i}^{2})+O(1\/r_{i}^{3})$. Therefore we have\n\\begin{equation}\\label{THI}\n\\frac{2}{r_{i}}+\\frac{2m}{r_{i}(r_{i}-2m)}+O(1\/r_{i}^{3})\\leq  \\frac{2}{{\\tt s}_{\\Gamma_{i}}+\\frac{A}{8\\pi m}} + \\frac{2{\\mathcal{M}}(q_{i})}{({\\tt s}_{\\Gamma_{i}}+\\frac{A}{8\\pi m})^{2}N^{2}(p_{i})}.\n\\end{equation}\n\n\\noindent We can arrange this better as\n\\begin{equation}\\label{THI2}\n\\frac{2({\\tt s}_{\\Gamma_{i}} +\\frac{A}{8\\pi m} -r_{i})}{r_{i}({\\tt s}_{\\Gamma_{i}}+\\frac{A}{8\\pi m})} +\\frac{2m}{r_{i}(r_{i}-2m)} - \\frac{2{\\mathcal{M}}(q_{i})}{({\\tt s}_{\\Gamma_{i}}+\\frac{A}{8\\pi m})^{2}N^{2}(p_{i})}\\leq O(1\/r_{i}^{3}).\n\\end{equation}\n\n\\noindent Secondly, from Proposition \\ref{MH} we have $\\lim {\\mathcal{M}}(q_{i})=|\\nabla N|_{H} (\\frac{A}{8\\pi m})^{2}=\\frac{A}{16\\pi m}$. Finally, we have $\\lim {\\tt s}(p_{i})-{\\tt s}_{\\Gamma_{i}}(p_{i})=0$ and from Proposition \\ref{CDLF} it is $\\lim r_{i}\/{\\tt s}_{\\Gamma_{i}}=1$. Multiplying equation (\\ref{THI2}) by ${\\tt s}_{\\Gamma_{i}}^{2}$, taking the limsup while using the facts described above gives finally\n\\begin{displaymath}\n\\bar{\\delta}(\\{p_{i}\\})=\\limsup {\\tt s}(p_{i})-r_{i}+2m\\leq m(1-P).\n\\end{displaymath}\n\n\\noindent as desired.\\hspace{\\stretch{1}}$\\Box$ \n\n\\vs\n\\noindent Using Corollary \\ref{CDC} and Proposition \\ref{PDC} we deduce the Penrose inequality.\n\n\\begin{Proposition} {\\rm{\\bf (The Penrose inequality).}}\\label{PIT}\nLet $(\\Sigma,g,N)$ be an asymptotically flat static solution with a regular and connected horizon $H$. Let $A$ be the area of $H$ and $m$ the ADM mass of the solution. Then\n\\begin{equation}\\label{PENR}\nA\\leq 16\\pi m^{2}.\n\\end{equation}\n\\end{Proposition}\n \n\\subsection{The opposite Penrose inequality.}\\label{OSPI}\n\nIn this Section we prove the {\\it opposite Penrose inequality} namely that $A\\geq 16\\pi m^{2}$. The proof will follow after carefully studying the behavior of the quotient $\\hat{{\\tt s}}\/{\\tt s}$ at the  singularity of ${\\tt g}$, namely the (unique) horizon $H$, and using then the distance comparison in Theorem \\ref{DC}. We will denote by $\\kappa$ the Gaussian curvature of the two-metric on $H$ inherited from $g$. \n\n\\begin{Proposition}\\label{EMB}\nLet $(\\Sigma, g, N)$ be an asymptotically flat static solution with regular and connected horizon. Consider a $g$-geodesic $\\gamma$ starting perpendicularly from $H$ at $q$, and parameterized with respect to the $g$-arc length of $\\gamma$ from $q$, $\\tau$. Define $\\hat{\\hat{{\\tt s}}}(\\gamma(\\tau))=\\int_{0}^{\\tau}N(\\gamma(\\tau))d\\tau$. Then we have\n\\begin{equation}\\label{LIM0}\n\\frac{d}{d \\hat{\\hat{{\\tt s}}}}\\frac{\\hat{{\\tt s}}}{\\hat{\\hat{{\\tt s}}}}\\bigg|_{q}=8m(\\frac{4\\pi m}{A})^{2} - 2m \\kappa\\ \\bigg|_{q}.\n\\end{equation}\n\\end{Proposition}\n\n\\noindent {\\it Proof:}\n\n\\vs\nNote that, as is written in the statement of the Proposition, we will work in the natural representation $(\\Sigma,g,N)$ of the static solution.  \n\nNow first we note that $d\\hat{\\hat{{\\tt s}}}(\\tau)\/d\\tau =N(\\alpha(\\tau))$. Derivatives with respect to $\\tau$ will be denoted by a prima, i.e. $f'(\\alpha(\\tau))'=d f(\\alpha(\\tau))\/d\\tau$. We compute (when $\\tau\\neq 0$)\n\\begin{equation}\\label{LIM}\n\\frac{d}{d\\hat{\\hat{{\\tt s}}}} \\frac{\\hat{{\\tt s}}}{\\hat{\\hat{{\\tt s}}}}=\\frac{2m((2\\frac{N'}{1-N^{2}}+2\\frac{N^{2}N'}{(1-N^{2})^{2}})\\hat{\\hat{{\\tt s}}}-2m\\frac{N^{2}}{(1-N^{2})})}{\\hat{\\hat{{\\tt s}}}^{2}}.\n\\end{equation}\n\n\\noindent We want to calculate now the limit of this expression when $\\tau\\rightarrow 0$.  We will separate the right hand side of (\\ref{LIM}) into two terms and calculate the limit for each one of them separately. The first limit we will calculate is\n\\begin{equation}\\label{LIMIN}\n\\lim_{\\tau\\rightarrow 0} \\frac{4mN^{2} N'}{(1-N^{2})^{2}\\hat{\\hat{{\\tt s}}}}=4m|\\nabla N|_{H} \\lim_{\\tau\\rightarrow 0} \\frac{N^{2}}{\\hat{\\hat{{\\tt s}}}},\n\\end{equation}\n\n\\noindent which arises from the middle term on the right hand side of equation (\\ref{LIM}). The right hand side of (\\ref{LIMIN}) was obtained using that $N'(\\tau)\\rightarrow |\\nabla N|_{H}$ and $(1-N^{2})^{2}\\rightarrow 1$. We calculate now the limit on the right hand side of (\\ref{LIMIN}) using L'H${\\rm\\hat{o}}$pital rule and we have\n\\begin{displaymath}\n\\lim_{\\tau\\rightarrow 0} \\frac{N^{2}}{\\hat{\\hat{{\\tt s}}}}=\\lim_{\\tau\\rightarrow 0} 2 N'=2|\\nabla N|_{H}.\n\\end{displaymath}\n\n\\noindent Thus we get \n\\begin{equation}\\label{1L}\n\\lim_{\\tau\\rightarrow 0} \\frac{4mN^{2} N'}{(1-N^{2})^{2}\\hat{\\hat{{\\tt s}}}}=8m (|\\nabla N|^{2}_{H})=8m(\\frac{4\\pi m}{A})^{2}.\n\\end{equation}\n\n\\noindent The second limit that we will calculate is\n\\begin{equation}\\label{2L}\n\\lim_{\\tau\\rightarrow 0} \\frac{2m}{1-N^{2}} \\frac{(2 N'\\hat{\\hat{{\\tt s}}}-N^{2})}{\\hat{\\hat{{\\tt s}}}^{2}}= 2m \\lim_{\\tau\\rightarrow 0}\\frac{(2 N'\\hat{\\hat{{\\tt s}}}-N^{2})}{\\hat{\\hat{{\\tt s}}}^{2}}.\n\\end{equation}  \n\n\\noindent which arises from the combination of the first and third term on the right hand side of (\\ref{LIM}). Again, to obtain the right hand side of (\\ref{2L}), we use the fact that the factor $2m\/(1-N^{2})$ would be, in the limit, $2m$. We calculate the limit on the right hand side of (\\ref{2L}) by L'H${\\rm{\\hat{o}}}$pital rule, and obtain\n\\begin{equation}\\label{3L}\n2m \\lim_{\\tau\\rightarrow 0} \\frac{ (2N'-2N'+2\\hat{\\hat{{\\tt s}}} \\frac{N''}{N})}{2\\hat{\\hat{{\\tt s}}}}=2m\\ Ric(n,n),\n\\end{equation}\n\n\\noindent where $n=\\alpha'(0)$ is the outward $g$-unit normal vector to $H$ at $\\alpha(0)$. To obtain the right hand side above we used the static equation (\\ref{SE2}), namely $N''(\\alpha(0))=Ric(\\alpha'(0),\\alpha'(0))N(\\alpha(0))$ (note that $\\alpha(\\tau)$ is a $g$-geodesic). \n\nRecall now the structure equation $2\\kappa (q) + |\\Theta|^{2}(q)-\\theta^{2}(q)=R(q)-2Ric(n(q),n(q))$, where $q$ is a point in $H$. Again, $n$ is the outward $g$-unit normal vector to $H$ at $q$. $\\Theta(q)$ and $\\theta(q)$ are the second fundamental forms of $H$, calculated using $g$, and evaluated at $q$. For a regular horizon we know that $\\Theta=0$, $\\theta=0$. $R$ and $Ric$ are the scalar and Ricci curvatures of $g$ respectively. For a static solution $(\\Sigma,g,N)$ it is $R=0$ everywhere. $\\kappa$, as said above is the Gaussian curvature of $H$ with the two-metric inherited from $g$. Thus, from the structure equation we get that for all $q$ in $H$ we have $\\kappa(q)=-Ric(n,n)$. Using this fact in (\\ref{3L}) and combining (\\ref{3L}) and (\\ref{1L}) to complete the limit (\\ref{LIM}), we obtain (\\ref{LIM0}).\\hspace{\\stretch{1}}$\\Box$\n\n\\begin{Proposition}\\label{EMB2}\nLet $(\\Sigma, {\\tt g}, N)$ be an asymptotically flat static solution with regular and connected horizon. If there is a point $q$ at $H$ for which\n\\begin{equation}\\label{GC}\n \\kappa(q)< 4(\\frac{4\\pi m}{A})^{2}.\n\\end{equation}\n\n\\noindent then there is a point $p$ in $\\Sigma\\setminus H$ such that $\\hat{{\\tt s}}(p)\/{\\tt s}(p)>1\/P$, where $P$ is the Penrose quotient.\n\\end{Proposition}\n\n\\noindent {\\it Proof:}\n\n\\vs\nSuppose there is a point $q$ in $H$ for which inequality (\\ref{GC}) hods. By Proposition \\ref{EMB}, there is a $g$-geodesic emanating perpendicularly to $H$ for which \n\\begin{displaymath}\n\\frac{d}{d \\hat{\\hat{{\\tt s}}}}\\frac{\\hat{{\\tt s}}}{\\hat{\\hat{{\\tt s}}}}>0.\n\\end{displaymath}\n\n\\noindent Also applying L'h\\^opital rule we get\n\\begin{displaymath}\n\\lim_{\\tau\\rightarrow 0} \\frac{\\hat{{\\tt s}}}{\\hat{\\hat{{\\tt s}}}}=\\lim_{\\tau\\rightarrow 0} \\frac{\\frac{4mNN'}{(1-N^{2})^{2}}}{N}=4m|\\nabla N|_{H}=\\frac{1}{P}.\n\\end{displaymath}\n\n\\noindent Therefore we have $\\hat{{\\tt s}}(\\gamma(\\tau))\/\\hat{\\hat{{\\tt s}}}(\\gamma(\\tau))>1\/P$ for $\\tau$ small. Now we observe that $\\hat{\\hat{{\\tt s}}}(\\gamma(\\tau))\\geq {\\tt s}(\\gamma(\\tau))$ because ${\\tt s}$ is the ${\\tt g}$-distance function to $H$ and $\\hat{\\hat{{\\tt s}}}(\\gamma(\\tau))$ is the ${\\tt g}$-length of $\\gamma$ between $\\gamma(0)$ and $\\gamma(\\tau)$. Thus, for $\\tau$ small we have\n\\begin{displaymath}\n\\frac{\\hat{{\\tt s}}(\\gamma(\\tau))}{{\\tt s}(\\gamma(\\tau))} = \\frac{\\hat{{\\tt s}}(\\gamma(\\tau))}{\\hat{\\hat{{\\tt s}}}(\\gamma(\\tau))}\\frac{\\hat{\\hat{{\\tt s}}}(\\gamma(\\tau))}{{\\tt s}(\\gamma(\\tau))} \\geq \n\\frac{\\hat{{\\tt s}}(\\gamma(\\tau))}{\\hat{\\hat{{\\tt s}}}(\\gamma(\\tau))}>\\frac{1}{P}.\n\\end{displaymath}   \n\\hspace{\\stretch{1}}$\\Box$\n\n\\begin{Corollary}\\label{CORIPI} Let $(\\Sigma,g,N)$ be an asymptotically flat static solution with regular and connected horizon $H$. Then, $H$ is homeomorphic to a two-sphere and the inverse Penrose inequality holds, $A\\geq 16\\pi m^{2}$. Moreover if the Penrose inequality holds, namely $A\\leq 16\\pi m^{2}$, then $\\kappa=4\\pi\/A$ and the horizon is round.\n\\end{Corollary}\n\n\\noindent {\\it Proof:}\n\n\\vs\n\tBy Proposition \\ref{EMB2} if there is a point $q$ in $H$ for which $\\kappa(q)<4(4\\pi m\/A)^{2}$ then there is point $p$ in $\\Sigma\\setminus H$ such that $\\hat{s}(p)\/s(p)>1\/P$ but this contradicts the distance comparison of Theorem \\ref{DC}. Therefore $\\kappa\\geq 4(4\\pi m\/A)^{2}$ and, by Gauss-Bonnet, $H$ must be homeomorphic to a two sphere. Moreover\n\\begin{displaymath}\n\\int_{H}\\kappa dA=4\\pi\\geq 4(\\frac{4\\pi m}{A})^{2}.\n\\end{displaymath}\n\n\\noindent Thus\n\\begin{displaymath}\nA\\geq 16\\pi m^{2},\n\\end{displaymath}\n\n\\noindent which finishes the first part of the claim. Suppose now that $A\\leq 16\\pi m^{2}$ then, as $\\kappa\\geq 4(4\\pi m\/A)^{2}$ we must have $k=4(4\\pi m\/A)^{2}=4\\pi\/A$ which finishes the claim.\\hspace{\\stretch{1}}$\\Box$ \n\n\\subsection{The uniqueness of the Schwarzschild solution.}\\label{USS}\n\n\\subsubsection{Further properties of the coordinate-distance lag.}\\label{FPCDL}\n\nThe proof of the uniqueness of the Schwarschild solutions does not follows directly in our setting from the equality $A=16\\pi m^{2}$. Indeed it is required first to prove that for any divergence sequence $\\{p_{i}\\}$ the associated coordinate-distance lag $\\bar{\\delta}(\\{p_{i}\\})$ is zero. We advocate now to prove this intermediate step. We need two preliminary Propositions. We start showing that $|\\nabla \\hat{\\st}|\\leq 1$. \n\n\\begin{Proposition}\\label{GRADUN} Let $(\\Sigma,{\\tt g},\\ln N)$ be an asymptotically flat static solution with regular an connected horizon. Then, $|\\nabla \\hat{{\\tt s}}|_{{\\tt g}}\\leq 1$. \n\\end{Proposition}\n\n\\vs\n\\noindent {\\it Proof:}\n\n\tWe observe first that $\\lim_{{\\tt s}(p)\\rightarrow \\infty}|\\nabla \\hat{{\\tt s}}|_{{\\tt g}}(p)=1$. But we also have $\\lim_{{\\tt s}(p)\\rightarrow 0}|\\nabla \\hat{{\\tt s}}|_{{\\tt g}}=1$. To see this last claim we compute\n\\begin{displaymath}\n|\\nabla \\hat{{\\tt s}}|_{{\\tt g}}(p)=\\frac{4m}{(1-N^{2}(p))^{2}}|\\nabla N(p)|_{g}\\rightarrow 4m |\\nabla N|_{H}.\n\\end{displaymath}\n\t\n\\noindent But we already know from Corollary \\ref{CORIPI} that $P=1$ and thus $|\\nabla N|_{H}=4\\pi m\/A=1\/4m$. The claim follows. \n\t\n\tWe show now that there cannot exist a point $p$ in $\\Sigma\\setminus H$ for which $|\\nabla \\hat{{\\tt s}}|(p)>1$. \nWe will assume without loss of generality that $m=1$. The assumption simplifies the writing. Define\n\\begin{displaymath}\n\\hat{{\\tt s}}_{\\alpha}=\\frac{1}{1-N^{2\\alpha}}-1,\n\\end{displaymath}\n\n\\noindent and thus \n\\begin{displaymath}\nN^{2\\alpha}=\\frac{\\hat{{\\tt s}}_{\\alpha}}{\\hat{{\\tt s}}_{\\alpha}+1}.\n\\end{displaymath}\n\n\\noindent Then we compute\n\\begin{displaymath}\n2\\alpha N^{2\\alpha-1}\\nabla N=\\frac{1}{(\\hat{\\st}_{\\alpha}+1)^{2}}\\nabla \\hat{\\st}_{\\alpha},\n\\end{displaymath}\n\n\\noindent and thus\n\\begin{displaymath}\n\\frac{\\nabla N}{N}=\\frac{1}{2\\alpha \\hat{\\st}_{\\alpha}(\\hat{\\st}_{\\alpha}+1)}\\nabla \\hat{\\st}_{\\alpha}.\n\\end{displaymath}\n\n\\noindent But $\\Delta \\ln N=0$ and then $\\nabla (1\/(\\hat{\\st}_{\\alpha}(\\hat{\\st}_{\\alpha}+1))\\nabla \\hat{\\st}_{\\alpha})=0$ which can be written as\n\\begin{equation}\\label{LAPSA}\n\\Delta \\hat{\\st}_{\\alpha}=\\frac{2\\hat{\\st}_{\\alpha}+1}{\\hat{\\st}_{\\alpha}(\\hat{\\st}_{\\alpha}+1)}|\\nabla \\hat{\\st}_{\\alpha}|^{2}.\n\\end{equation}\n\n\\noindent The interesting thing about this expression is that it does not depend explicitly on $\\alpha$. We note too that we have \n\\begin{equation}\\label{STAGRAD}\n<\\frac{\\nabla N}{N},\\nabla \\hat{\\st}_{\\alpha}>=\\frac{1}{2\\alpha \\hat{\\st}_{\\alpha}(\\hat{\\st}_{\\alpha}+1)}|\\nabla \\hat{\\st}_{\\alpha}|^{2}.\n\\end{equation}\n\n\\noindent The crucial and obvious observation about the family $\\{\\hat{\\st}_{\\alpha}\\}$ is that given any open set $\\Omega$ of compact closure $\\bar{\\Omega}\\subset \\Sigma\\setminus H$ then $\\hat{\\st}_{\\alpha}$ converges uniformly in $C^{2}$ to ${\\tt s}$ over $\\bar{\\Omega}$ as $\\alpha\\rightarrow 1$. Thus it follows from the limits of ${\\tt s}$ at $H$ and infinity observed at the beginning that if $\\max\\{|\\nabla {\\tt s}|(q),q\\in \\Sigma\\}>1$ then there is an $\\epsilon>0$ such that for every $\\alpha$ with $|\\alpha -1|<\\epsilon$ the function $|\\nabla \\hat{\\st}_{\\alpha}|$ posses at least one local maximum greater than one. For a given $\\alpha$ we will denote by $p_{\\alpha}$ a point at which a local maximum of $\\hat{\\st}_{\\alpha}$ greater than one takes place.  \n\nWe will use Weitzenb\\\"ock's formula\n\\begin{equation}\\label{WEF}\n\\frac{1}{2}\\Delta |\\nabla \\hat{\\st}_{\\alpha}|^{2}=|\\nabla \\nabla \\hat{\\st}_{\\alpha}|^{2}+<\\nabla \\Delta \\hat{\\st}_{\\alpha},\\nabla \\hat{\\st}_{\\alpha}>+2<\\frac{\\nabla N}{N},\\nabla \\hat{\\st}_{\\alpha}>^{2},\n\\end{equation}\n\n\\noindent and we will use it evaluated at $p_{\\alpha}$. We note first that for every vector $w\\in T_{p_{\\alpha}}\\Sigma$ we have $<\\nabla_{w}\\nabla \\hat{\\st}_{\\alpha},\\nabla \\hat{\\st}_{\\alpha}>=0$. Because of this we have $|\\nabla\\nabla\\hat{\\st}_{\\alpha}|^{2}=|\\nabla\\nabla \\hat{\\st}_{\\alpha}|_{T_{p_{\\alpha}}\\Sigma}^{2}=|\\nabla\\nabla \\hat{\\st}_{\\alpha}|_{\\nabla\\hat{\\st}_{\\alpha}(p_{\\alpha})^{\\perp}}$ where $\\nabla\\hat{\\st}_{\\alpha}(p_{\\alpha})^{\\perp}$ is the perpendicular subspace to $\\nabla \\hat{\\st}_{\\alpha}$ in $T_{p_{\\alpha}}\\Sigma$. Thus we have \n\\begin{displaymath}\n|\\nabla \\nabla \\hat{\\st}_{\\alpha}|^{2}(p_{\\alpha})\\geq \\frac{1}{2} tr_{\\nabla \\hat{\\st}_{\\alpha}(p_{\\alpha})^{\\perp}}\\nabla \\nabla\\hat{\\st}_{\\alpha}=\\frac{1}{2}\\Delta \\hat{\\st}_{\\alpha} (p_{\\alpha}).\n\\end{displaymath}\n\n\\noindent This expression will be used in the first term on the right hand side of equation (\\ref{WEF}). For the second instead we note from equation (\\ref{LAPSA}) that\n\\begin{displaymath}\n\\nabla \\Delta \\hat{\\st}_{\\alpha} \\bigg|_{p_{\\alpha}}=-(\\frac{1}{\\hat{\\st}_{\\alpha}^{2}}+\\frac{1}{(\\hat{\\st}_{\\alpha}+1)^{2}})|\\nabla \\hat{\\st}_{\\alpha}|^{2}\\bigg|_{p_{\\alpha}}. \n\\end{displaymath}\n\n\\noindent For the third term on the right hand side of equation (\\ref{WEF}) we will use equation (\\ref{STAGRAD}). All together gives for equation (\\ref{WEF}) the expression\n\\begin{displaymath}\n0\\geq \\frac{1}{2}\\Delta |\\nabla \\hat{\\st}_{\\alpha}|^{2}\\bigg|_{p_{\\alpha}}\\geq  |\\nabla \\hat{\\st}_{\\alpha}|^{2}(\\frac{(2\\hat{\\st}_{\\alpha}+1)^{2}}{2(\\hat{\\st}_{\\alpha}^{2}(\\hat{\\st}_{\\alpha}+1)^{2}}-\\frac{\\hat{\\st}_{\\alpha}^{2}+(\\hat{\\st}_{\\alpha}+1)^{2}}{\\hat{\\st}_{\\alpha}^{2}(\\hat{\\st}_{\\alpha}+1)^{2}}+\\frac{2}{4\\alpha^{2}}\\frac{1}{\\hat{\\st}_{\\alpha}^{2}(\\hat{\\st}_{\\alpha}+1)^{2}})\\bigg|_{p_{\\alpha}}.\n\\end{displaymath}\n\n\\noindent Further expanding the term in parenthesis we obtain\n\\begin{displaymath}\n0\\geq \\frac{1}{2}\\Delta |\\nabla \\hat{\\st}_{\\alpha}|^{2}\\bigg|_{p_{\\alpha}}\\geq \\frac{|\\nabla \\hat{\\st}_{\\alpha}|^{2}}{2\\hat{\\st}_{\\alpha}^{2}(\\hat{\\st}_{\\alpha}+1)^{2}}(-1+\\frac{1}{\\alpha})\\bigg|_{p_{\\alpha}}.\n\\end{displaymath}\n\n\\noindent Choosing $\\alpha$ such that $1-\\epsilon <\\alpha<1$ we get a contradiction. This finishes the proof of the Proposition. \\hspace{\\stretch{1}}$\\Box$\n\n\nDefine now $\\delta={\\tt s}-\\hat{{\\tt s}}$. We will study $\\delta$, and it will be shown that it has asymptotically positive Laplacian (in a barer sense). \n\n\\begin{Proposition} Let $(\\Sigma,{\\tt g},\\ln N)$ be an asymptotically flat static solution with regular and connected horizon $H$. The Laplacian of $\\delta$ has the following asymptotic expression\n\\begin{displaymath}\n\\Delta \\delta\\leq \\frac{-\\delta}{(s+2m)^{2}}+O(\\frac{1}{s^{3}}),\n\\end{displaymath}\n\n\\noindent in the barer sense.\n\\end{Proposition}\n\nNote that $\\delta\\geq 0$. However note too that because there are sequences $\\{p_{i}\\}$ for which $\\delta(p_{i})\\rightarrow 0$, it cannot be said that $\\Delta \\delta$ becomes negative outside a sufficiently big compact set. The asymptotic expression is however still valid.\n\n\\noindent {\\it Proof:}\n\n\\vs\nRecall first the expression for $\\Delta \\hat{\\st}$ in equation (\\ref{SBF}). We find first the asymptotic expression for $|\\nabla \\hat{\\st}|^{2}$. But observing that $\\hat{\\st} = 2m(\\frac{1}{1-N^{2}}-1)$ it is easily deduced from the asymptotic expression of $N$ that $\\nabla \\hat{\\st}=\\nabla r+O(1\/r^{2})$. Thus $|\\nabla \\hat{\\st}|^{2}=1+(1\/r^{2})=1+O(1\/s^{2})$. \n\nNow subtract to the expression (\\ref{SL}) with $P=1$ and $|\\nabla {\\tt s}|^{2}=1$, the expression (\\ref{SBF}). That gives\n\\begin{displaymath}\n\\Delta \\delta\\leq \\frac{2}{{\\tt s}+2m}(1+\\frac{m}{{\\tt s}+2m})-\\frac{2}{\\hat{\\st}+2m}(1+\\frac{m}{\\hat{\\st}+2m})+O(\\frac{1}{{\\tt s}^{3}})=\n\\end{displaymath}\n\\begin{displaymath}\n=\\frac{-2\\delta}{({\\tt s}+2m)(\\hat{\\st}+2m)}+2m\\frac{(\\hat{\\st}^{2}-{\\tt s}^{2})}{({\\tt s}+2m)^{2}(\\hat{\\st}^{2}+2m)^{2}}+O(\\frac{1}{{\\tt s}^{3}}).\n\\end{displaymath}\n\n\\noindent Thus \n\\begin{displaymath}\n\\Delta \\delta\\leq \\frac{-\\delta}{({\\tt s}+2m)^{2}}+O(\\frac{1}{{\\tt s}^{3}}).\n\\end{displaymath}\n\n\\noindent as claimed.\\hspace{\\stretch{1}}$\\Box$\n \n\\vs \nWe prove now a crucial property of $\\delta$, namely that  it is Lipschitz ``at large scales\". To explain the concept we need to introduce some terminology. Let $\\{(r,\\theta,\\varphi)\\}$ a be a coordinate system as in \nProposition \\ref{PBS2}. Let $D$ be the annulus in $\\field{R}^{3}$, $D=\\{(r,\\theta,\\varphi),1\\leq r\\leq2\\}$. For any $\\lambda>0$ sufficiently small consider the map from $D$ into $\\Sigma$ given by $\\bar{x}\\rightarrow \\bar{x}\/\\lambda$. Denote by $\\delta_{\\lambda}$ the pull-back of $\\delta$ to $D$, namely $\\delta_{\\lambda}(\\bar{x})=\\delta(\\bar{x}\/\\lambda)$. \nLet $\\bar{x}_{1}$ and $\\bar{x}_{2}$ be two points in $D$. Denote by $\\phi(\\bar{x}_{s},\\bar{x}_{2})$ the angle formed by $\\bar{x}_{1}$ and $\\bar{x}_{2}$, namely $<\\bar{x}_{1},\\bar{x}_{2}>=|\\bar{x}_{1}||\\bar{x}_{2}|\\cos \\phi(\\bar{x}_{1},\\bar{x}_{2})$. We would like to show that there is $\\lambda_{0}>0$ and $K>0$ such that $\\delta_{\\lambda}$ is Lipschitz with constant $K$ for any $0<\\lambda<\\lambda_{0}$. The next Proposition explains this property and two further that will also be needed later. It is perhaps the most technical, but otherwise straightforward Proposition of the article.\n\\begin{Proposition} Let $\\delta={\\tt s}-\\hat{\\st}$. Then\n\\begin{enumerate} \n\\item There exists $K>0$ and $\\lambda_{0}>0$ such that for any $\\bar{x}_{1},\\bar{x}_{2}$ in $D$ and $0<\\lambda<\\lambda_{0}$ we have\n\\begin{displaymath}\n|\\delta_{\\lambda}(\\bar{x}_{1})-\\delta_{\\lambda}(\\bar{x}_{2})|\\leq K |\\bar{x}_{1}-\\bar{x}_{2}|,\n\\end{displaymath}\n \n\\item Let $\\bar{x}_{1}$ and $\\bar{x}_{2}$ be two points in $D$ belonging to the same radial line, namely $\\bar{x}_{1}=\\beta \\bar{x}_{2}$. Then for any sequence $\\{\\lambda_{i}\\}\\downarrow 0$ we have\n$|\\delta_{\\lambda_{1}}(\\bar{x}_{1})-\\delta_{\\lambda_{i}}(\\bar{x}_{2})|\\rightarrow 0$.\n\\end{enumerate}\n\\end{Proposition}\n\n\\noindent {\\it Proof:}\n\n\\vs\nIn $\\Sigma$ consider a coordinate sphere $S_{r_{0}}=\\{\\bar{x}\/ r(\\bar{x})=r_{0}\\}$ (where $\\{\\bar{x}\\}$ is a coordinate system as in Proposition \\ref{PBS2}). The distance function from $S_{r_{0}}$ to $H$ is Lipschitz, say with constant $K_{1}$, namely for any $q_{0}$, $q_{1}$ in $S_{r_{0}}$ we have\n$|{\\tt s}(q_{0})-{\\tt s}(q_{1})|\\leq K_{1} |\\phi(q_{0},q_{1})|$. \n\nLet now $\\bar{x}_{1}$ be a point in $D$. Let $\\lambda$ such that $|\\bar{x}_{1}|\/\\lambda >>r_{0}$. Denote $p_{1}=\\bar{x}_{1}\/\\lambda$. Let $\\gamma_{1}$ be the length minimizing geodesic joining $\\bar{x}_{1}\/\\lambda$ to $H$. Let $q_{1}$ be the point of intersection of $\\gamma_{1}$ with $S_{r_{0}}$. Consider a rotation of angle $\\phi_{0}$ in $\\field{R}^{3}$, denote it by $R_{\\phi_{0}}$. Also denote by $p_{2}=R_{\\phi_{0}}(p_{1})$, $\\gamma_{2}=R_{\\phi_{0}}(\\gamma_{1})$ and $q_{2}=R_{\\phi_{0}}(q_{1})$. Let $l_{1}$ be the length of $\\gamma_{1}$ between $p_{1}$ and $q_{1}$ and let $l_{2}$ be the length between $p_{2}$ and $q_{2}$ of $\\gamma_{2}$. \n\nWe will show first that there is a constant $K_{2}>0$ independent on $\\lambda$ such that $|l_{1}-l_{2}|\\leq K_{2} |\\phi_{0}|$. Note that in the coordinate system $\\{\\bar{x}\\}$ we have ${\\tt g}={\\tt g}_{S}+O(1\/r^{3})$. Suppose $\\gamma_{1}$ is parameterized with respect to the arc-length, $\\bar{{\\tt s}}$, provided by the Schwarzschild metric ${\\tt g}_{S}$. Let $l(\\phi)=l(R_{\\phi}(\\gamma_{1}))$, where $0<\\phi<\\phi_{0}$. Then we have  \n\\begin{equation}\\label{DERL}\n|\\partial_{\\phi}l|=|\\int_{\\bar{{\\tt s}}_{0}=0}^{\\bar{{\\tt s}}_{1}}\\frac{{\\tt g}(\\nabla_{\\partial_{\\phi}}\\gamma',\\gamma')}{{\\tt g}(\\gamma',\\gamma')^{\\frac{1}{2}}}d\\bar{{\\tt s}}|. \n\\end{equation} \n\n\\noindent Moreover \n\\begin{displaymath}\n{\\tt g}(\\nabla_{\\partial_{\\phi}}\\gamma',\\gamma')={\\tt g}((\\nabla_{\\partial_{\\phi}}-\\nabla^{S}_{\\partial_{\\phi}})\\gamma',\\gamma')+({\\tt g}-{\\tt g}_{S})(\\nabla^{S}_{\\partial_{\\phi}}\\gamma',\\gamma')+{\\tt g}_{S}(\\nabla^{S}_{\\partial_{\\phi}}\\gamma',\\gamma').\n\\end{displaymath}\n\n\\noindent We note now that the last term on the right hand side of the previous equation is zero, and the first two terms on the right hand side are $O(1\/\\bar{{\\tt s}}^{2})$. Using this in equation (\\ref{DERL}) we get that $|l_{1}-l_{2}|\\leq K_{2}|\\phi_{0}|$ as desired.  \n\nWe have now\n\\begin{displaymath}\n{\\tt s}(p_{2})\\leq l_{2}+{\\tt s}(q_{2})\\leq l_{1}+{\\tt s}(q_{1})+K_{1}|\\phi_{0}|+K_{2}|\\phi_{0}|={\\tt s}(p_{1})+(K_{1}+K_{2})|\\phi_{0}|.\n\\end{displaymath}\n\n\\noindent Because $p_{1}$ and $\\phi$ are arbitrary we have\n\\begin{displaymath}\n|{\\tt s}(p_{1})-{\\tt s}(p_{2})|\\leq K|\\phi_{0}|.\n\\end{displaymath}\n\n\\noindent Thus for any $\\bar{x}_{1}$ and $\\bar{x}_{2}$ in $D$ of equal norm, $|\\bar{x}_{1}|=|\\bar{x}_{2}|$, and $\\lambda$ (sufficiently small), we have \n\\begin{equation}\\label{IIU}\n|\\delta_{\\lambda}(\\bar{x}_{1})-\\delta_{\\lambda}(\\bar{x}_{2})|=|{\\tt s}(\\frac{\\bar{x}_{1}}{\\lambda})-\\frac{|\\bar{x}_{1}|}{\\lambda}+2m-{\\tt s}(\\frac{\\bar{x}_{2}}{\\lambda})+\\frac{|\\bar{x}_{2}|}{\\lambda}-2m|=|{\\tt s}(\\frac{\\bar{x}_{1}}{\\lambda})-{\\tt s}(\\frac{\\bar{x}_{2}}{\\lambda})|\\leq K|\\phi(\\bar{x}_{1},\\bar{x}_{2})|.\n\\end{equation}\n\nWe continue with an observation. Recall that the Ricci curvature of ${\\tt g}$ decays, in $r$, as $O(1\/r^{3})$ (in facts it decays as $1\/r^{4}$). Consider the annulus $D_{\\lambda}=\\{\\bar{x},\\ \\lambda^{1\/12}\\leq |\\bar{x}|\\leq 2\\}$ and consider the map from $D_{\\lambda}$ into $\\Sigma$ given by $\\bar{x}\\rightarrow \\bar{x}\/\\lambda$. Let ${\\tt g}_{\\lambda}$ be the pull-back of the metric ${\\tt g}$ under this map. The from the fact that $|{\\tt Ric}|$ decays as $O(1\/r^{3})$ we get $\\sup\\{|{\\tt Ric}_{{\\tt g}_{\\lambda}}(\\bar{x})|_{{\\tt g}_{\\lambda}}\/ \\bar{x}\\in D_{\\lambda}\\}=O(\\lambda^{\\frac{1}{4}})$. From this it follows that, as $\\lambda$ tends to zero, and therefore as $D_{\\lambda}$ tends to the closed ball of radius two minus the origin, the metrics ${\\tt g}_{\\lambda}$ converge in $C^{1,\\beta}$ (for any $0<\\beta<1$) to the flat metric over any fixed annulus $D_{\\lambda_{1}}$, $0<\\lambda_{1}<2$. Thus for any $\\bar{x}\\in D$ and sequence $\\{\\lambda_{i}\\}\\downarrow 0$, length minimizing geodesics, $\\gamma_{p}$, joining $p=\\bar{x}\/\\lambda$ to $H$ converge in $C^{1}$ over any $D_{\\lambda_{1}}$ to the radial line  passing through $\\bar{x}$. \n\nWhat we would like to know now is the ``rate\" at which the geodesics approach the radial lines. More precisely, we will study the ${\\tt g}_{S}$-angle $\\xi$, formed by $\\partial_{r}$ and $\\gamma'$ at any point along $\\gamma$. To this respect we proceed as follows. Consider the rotational killing fields $X$ of the Schwarzschild solution. For every $X$, we have $|X|_{{\\tt g}}=r(1+O(1\/r))$. Given one of the $X$'s, we compute, along the geodesic $\\gamma_{p}$ (again $p=\\bar{x}\/\\lambda$)\n\\begin{displaymath}\n{\\tt g}(\\gamma',X)'={\\tt g}(\\gamma',\\nabla_{\\gamma'}X)={\\tt g}(\\gamma',(\\nabla_{\\gamma'}-\\nabla^{S}_{\\gamma'}) X)+{\\tt g}_{S}(\\gamma',\\nabla^{S}_{\\gamma'}X)+({\\tt g}-{\\tt g}_{S})(\\gamma',\\nabla^{S}_{\\gamma'} X).\n\\end{displaymath}     \n\n\\noindent The second term on the right hand side of the previous equation is zero, while the other two are of the order $O(1\/r^{2})=O(1\/{\\tt s}^{2})$. Let $q$ be the first point where $\\gamma_{p}$ reaches the radial sphere $S_{r_{0}}$ ($r_{0}$ is fixed) and let $p_{1}$ be any intermediate point between $p$ and $q$. Integrate now ${\\tt g}(\\gamma',X)'$ (with respect to the ${\\tt g}$ arc-length, ${\\tt s}$) between ${\\tt s}(p_{1})$ and the value of ${\\tt s}(q)$ using the estimate we have found before for ${\\tt g}(\\gamma',X)$\nto get\n\\begin{displaymath}\n|{\\tt g}(\\gamma',X)(p_{1})-{\\tt g}(\\gamma',X)(q)|\\leq c_{1},  \n\\end{displaymath}\n\n\\noindent where $c_{1}$ is a constant independent on $p_{1}$ and $q$. Note that this inequality is valid for any rotational Killing field $X$. Observing that rotational killing fields at $S_{r_{0}}$ have bounded norm, we get\n\\begin{displaymath}\n|{\\tt g}(\\gamma',X)(p_{1})|\\leq c_{2},\n\\end{displaymath}\n\n\\noindent where $c_{2}$ is a constant. Moreover\n\\begin{displaymath}\n{\\tt g}_{S}(\\gamma',X)={\\tt g}(\\gamma',X)+({\\tt g}_{S}-{\\tt g})(\\gamma',X)={\\tt g}(\\gamma',X)+O(1\/r).\n\\end{displaymath}\n\n\\noindent Thus we have\n\\begin{displaymath}\n|{\\tt g}_{S}(\\gamma',X)|\\leq c_{3},\n\\end{displaymath}\n\n\\noindent where $c_{3}$ is a constant. Pick now the rotational killing field $X$ which is collinear, at\n$p_{1}$, to the component of $\\gamma'$, ${\\tt g}_{S}$-perpendicular to $\\partial_{r}$. Let $\\xi$ be the ${\\tt g}_{S}$-angle formed by $\\partial_{r}$ and $\\gamma'$. We have \n\\begin{displaymath}\n|{\\tt g}_{S}(\\gamma',X)(p_{1})|=|X|_{{\\tt g}_{S}}(p_{1})||\\gamma'|_{{\\tt g}_{S}}|\\sin \\xi(p_{1})|\\leq c_{4},\n\\end{displaymath}\n\n\\noindent where $c_{4}$ is a constant. So we get\n\\begin{displaymath}\n|\\sin \\xi|\\leq \\frac{c_{5}}{r},\n\\end{displaymath}\n\n\\noindent where $c_{5}$ is a constant. We have\n\\begin{equation}\\label{RSI}\n\\frac{dr}{d{\\tt s}}={\\tt g}_{S}(\\nabla^{S} r,\\gamma')=1+O(1\/r^{2})=1+O(1\/{\\tt s}^{2}).\n\\end{equation}\n\n\\noindent We will use this inequality in what follows. Let $\\bar{x}_{1}$ be a point in $D$. Let $p_{1}=\\bar{x}_{1}\/\\lambda$ and let $\\gamma$ be a geodesic minimizing the length between $p_{1}$ and $H$. Let $p_{2}$ be a point in $\\gamma$ such that $p_{2}=\\bar{x}_{2}\/\\lambda$ with $\\bar{x}_{2}$ in $D$. Integrating (\\ref{RSI}) between ${\\tt s}(p_{1})$ and ${\\tt s}(p_{2})$ we get\n\\begin{displaymath}\nr(p_{1})-{\\tt s}(p_{1})=r(p_{2})-{\\tt s}(p_{2})+|\\bar{x}_{1}-\\bar{x}_{2}|O(\\lambda).\n\\end{displaymath}\n\n\\noindent Therefore\n\\begin{equation}\\label{IID}\n|\\delta_{\\lambda}(\\bar{x}_{1})-\\delta_{\\lambda}(\\bar{x}_{2})|=|\\bar{x}_{1}-\\bar{x}_{2}|O(\\lambda).\n\\end{equation}\n\nWe are ready to prove the Proposition. Let $\\bar{x}_{1}$ and $\\bar{x}_{2}$ be two points in $D$. Let $p_{1}=\\bar{x}_{1}\/\\lambda$ and $p_{2}=\\bar{x}_{2}\/\\lambda$. Let $p_{3}=\\bar{x}_{3}\/\\lambda$ be the point of intersection of the length minimizing geodesic joining $p_{1}$ to $H$ and the coordinate sphere $S_{|\\bar{x}_{2}\/\\lambda|}$. From (\\ref{IIU}) and (\\ref{IID}) we get\n\\begin{displaymath}\n|\\delta_{\\lambda}(\\bar{x}_{1})-\\delta_{\\lambda}(\\bar{x}_{2}|\\leq |\\delta_{\\lambda}(\\bar{x}_{1})-\\delta_{\\lambda}(\\bar{x}_{3})|+|\\delta_{\\lambda}(\\bar{x}_{3}-\\delta_{\\lambda}(x_{2})|\\leq |\\bar{x}_{1}-\\bar{x}_{3}|O(\\lambda)+K\\phi(\\bar{x}_{3},\\bar{x}_{2}).\n\\end{displaymath}\n\n\\noindent As $|\\bar{x}_{1}-\\bar{x}_{3}|\\leq c_{6} d_{D}(\\bar{x}_{1},\\bar{x}_{3})$, for some constant $c_{6}$, the {\\it item 1} of the Proposition follows. {\\it Item 2} follows from the fact that $O(\\lambda)\\rightarrow 0$, as $\\lambda\\rightarrow 0$.\\hspace{\\stretch{1}}$\\Box$\n\n\\vs\nThe following direct implication will be crucial for the discussion that follows.\n\n\\begin{Corollary}\nFor any sequence $\\{\\lambda_{i}\\}$ such that $\\lambda_{i}\\downarrow 0$, there exists a subsequence $\\{\\lambda_{i_{k}}\\}\\downarrow 0$ and a Lipschitz function $\\delta_{0}$ (depending on $\\{\\lambda_{i_{k}}\\}$) for which $\\delta_{\\lambda_{i_{k}}}$ converges uniformly to $\\delta_{0}$ on $D$. The function $\\delta_{0}$ is constant on radial lines.\n\\end{Corollary} \n\nWe would like now to prove that the coordinate-distance lag $\\bar{\\delta}(\\{p_{i}\\})$ of any divergent sequence $\\{p_{i}\\}$ is zero. Naturally, this is the same as saying that $\\delta$ converges uniformly to zero at infinity. If this is not the case, then it is simple to see, arguing by contradiction, that we would be in the following situation. There would exist $\\{\\lambda_{i}\\}$ with $\\lambda_{i}\\downarrow 0$ such that $\\delta_{\\lambda_{i}}$ converges uniformly to a Liptschitz function function $\\delta_{0}$ and there would exist points $x,y$ in $D$ for which $\\delta_{0}(x)=0$, $|x|=3\/2$ and $\\delta_{0}(y)>0$, $|y|=3\/2$ and $|x-y|<1\/2$. Assume we are in such situation. Define in $D$ the Euclidean balls $B_{x}=B(x,|x-y|)$ and $B_{y}=B(y,\\xi)$ where $\\xi$ is small enough to have $\\delta_{0}|_{B_{y}}>c_{1}>0$, where $c_{1}$ is a constant. Following ~\\cite{MR2243772} (pg. 258) we can find a function $h$ on $\\bar{B}_{x}$ such that\n\\begin{enumerate}\n\\item $h\\bigg|_{(\\partial(B_{x})\\setminus B_{y})}<c_{2}<0$, where $c_{2}$ is a constant,\n\\item $h(x)=0$,\n\\item $\\Delta_{g_{\\lambda_{i}}} h \\bigg|_{\\bar{B}_{x}}> c_{3}>0$, where $c_{3}$ is a constant and $g_{\\lambda_{i}}$ is the scaled metric $\\lambda_{i}^{2}g$.\n\\end{enumerate}\n\n\\noindent Note that the scaled metrics $\\lambda_{i}^{2}g$ converge (in $C^{\\infty}$) to the flat Euclidean metric. \nAs $\\delta_{\\lambda_{i}}$ converges uniformly to $\\delta_{0}$ we deduce that there is $\\mu_{0}>0$ such that for any $0<\\mu\\leq \\mu_{0}$ (and $i\\geq i_{0}(\\mu_{0})$) we have $(-\\delta_{\\lambda_{i}}+\\mu h )|_{\\partial B_{x}}<\\mu c_{4}<0$, where $c_{4}$ is a constant. We also have $\\lim (-\\delta_{\\lambda_{i}}(x)+\\mu h(x))\\rightarrow 0$. It follows that having chosen $i_{1}$ big enough, the function $-\\delta_{\\lambda_{i}}+\\mu h$, ($\\mu\\leq \\mu_{0}$), for $i\\geq i_{1}$ has a maximum on $B_{x}$. Denote it by $z_{i}$. If the function ${\\tt s}$ were to be smooth at $z_{i}\/\\lambda_{i}$ and therefore $-\\delta_{\\lambda_{i}}+\\mu h$ were smooth at $z_{i}$ then one would get a contradiction to the maximum principle, as for $i$ sufficiently big, one would have\n\\begin{displaymath}\n\\Delta_{g_{\\lambda_{i}}}(-\\tilde{\\delta}_{\\lambda_{i}}+\\mu h )(z_{i})\\geq  \\frac{\\mu c_{3}}{2}>0.\n\\end{displaymath}\n\nWe explain now how to use Proposition \\ref{LSH} to overcome the case when $z_{i}$ are not smooth points of ${\\tt s}$. One can replace ${\\tt s}$ by ${\\tt s}_{\\Gamma_{i}}$, for a suitable $\\{\\Gamma_{i}\\}\\downarrow 0$, in the expression $\\delta_{\\lambda_{i}}(x)=({\\tt s}-\\hat{\\st})(x\/\\lambda_{i})$ in such a way that the new expression $(-({\\tt s}_{\\Gamma_{i}}-\\hat{\\st})+\\mu h)(x\/\\lambda_{i}$), has a maximum $\\tilde{z}_{i}$ on $B_{x}$. Further, by Proposition \\ref{LSH} one can replace ${\\tt s}_{\\Gamma_{i}}$ by $\\tilde{{\\tt s}}_{\\Gamma_{i}}$ in such a way that the new expression $\\tilde{\\delta}_{\\lambda_{i}}(x)=(\\tilde{{\\tt s}}_{\\Gamma_{i}}-\\hat{\\st})(x\/\\lambda_{i})$  satisfies \n\\begin{enumerate}\n\\item $-\\tilde{\\delta}_{\\lambda_{i}}(x)=-(\\tilde{{\\tt s}}_{\\Gamma_{i}}-\\hat{\\st})(x\/\\lambda_{i})\\leq -({\\tt s}_{\\Gamma_{i}}-\\hat{\\st})(x\/\\lambda_{i}$),\n\\item $-\\tilde{\\delta}_{\\lambda_{i}}(\\tilde{z}_{i})=({\\tt s}_{\\Gamma_{i}}-\\hat{\\st})(\\tilde{z}_{i}\/\\lambda_{i}$), and thus $-\\tilde{\\delta}_{\\lambda_{i}}+\\mu h$ has a maximum at $\\tilde{z}_{i}$ on $B_{x}$.\n\\item $\\Delta_{g_{\\lambda_{i}}}( -\\tilde{\\delta}_{\\lambda_{i}}+\\mu h )(\\tilde{z}_{i})\\geq  \\frac{\\mu c_{3}}{2}.$\n\\end{enumerate}\n\n\\noindent These three facts now contradict the maximum principle.\\hspace{\\stretch{1}}$\\Box$\n\nWe have thus proved\n\n\\begin{Proposition}\\label{DELZ} Let $(\\Sigma,{\\tt g},\\ln N)$ be an asymptotically flat static solution with regular and connected horizon. Then for any divergent sequence $\\{p_{i}\\}$, the coordinate-distance lag $\\bar{\\delta}(\\{p_{i}\\})$ is zero. \n\\end{Proposition}\n   \n\\subsubsection{Area and volume comparison.}\\label{AVC}   \n   \n\\begin{Proposition}\\label{VOLZ} Let $(\\Sigma,{\\tt g},\\ln N)$ be an asymptotically flat static solution with regular an connected horizon. Consider a sequence $\\{\\Gamma_{i}\\}\\downarrow 0$. Let ${\\mathcal{F}}_{\\Gamma_{i}}$ be the congruence of length minimizing geodesics to $H_{\\Gamma_{i}}$. Then for every $L>0$ we have\n\\begin{displaymath}\nVol(\\cup_{\\gamma\\in {\\mathcal{F}}_{\\Gamma_{i}}, l(\\gamma)\\leq L}\\{\\gamma\\})\\rightarrow 0,\n\\end{displaymath}\n\n\\noindent as $\\Gamma_{i}\\downarrow 0$. Above $\\{\\gamma\\}$ means the set of points in $\\gamma$.\n\\end{Proposition}\n\n\\noindent {\\it Proof:}\n\n\\vs\nThe first goal to achieve is to make the monotonicity of ${\\mathcal{M}}$ to look like a {\\it comparison of areas} and consequently a {\\it comparison of volumes}. Let $\\{\\Gamma_{i}\\}\\downarrow 0$. Consider for each $\\Gamma_{i}$ the congruence ${\\mathcal{F}}_{\\Gamma_{i}}$ of length minimizing geodesics to $H_{\\Gamma_{i}}$. We will work outside the locus at all times. Let $dA$ be the element of area of the level sets of the congruence. Let ${\\tt s}_{\\Gamma_{i}}$ be the distance function to $H_{\\Gamma_{i}}$. Then \n\\begin{displaymath}\n\\theta=\\frac{1}{A}\\frac{dA}{d{\\tt s}_{\\Gamma_{i}}}.\n\\end{displaymath}\n\n\\noindent Let $\\gamma$ be a geodesic in ${\\mathcal{F}}_{\\Gamma_{i}}$. Consider ${\\mathcal{M}}_{a}$ with $a=2m$ over $\\gamma$. Denote by ${\\mathcal{M}}_{\\Gamma_{i}}$ the value of ${\\mathcal{M}}$ at the initial point of $\\gamma$ in $H_{\\Gamma_{i}}$. Then from the monotonicity of ${\\mathcal{M}}$ we have\n\\begin{displaymath}\n(\\frac{1}{2A}\\frac{dA}{d{\\tt s}_{\\Gamma_{i}}}({\\tt s}_{\\Gamma_{i}}+2m)^{2}-({\\tt s}_{\\Gamma_{i}}+2m))N^{2}\\leq {\\mathcal{M}}_{\\Gamma_{i}}.\n\\end{displaymath}    \n\n\\noindent Rearranging terms we get\n\\begin{displaymath}\n\\frac{d}{d{\\tt s}_{\\Gamma_{i}}}(\\frac{dA}{({\\tt s}_{\\Gamma_{i}}+2m)^{2})})\\leq \\frac{2{\\mathcal{M}}_{\\Gamma_{i}}}{N^{2}({\\tt s}_{\\Gamma_{i}}+2m)^{2}}dA.\n\\end{displaymath}\n\n\\noindent We thus get\n\\begin{displaymath}\n\\frac{d}{d{\\tt s}_{\\Gamma_{i}}} \\ln \\frac{dA}{({\\tt s}_{\\Gamma_{i}}+2m)^{2}}\\leq \\frac{2{\\mathcal{M}}_{\\Gamma_{i}}}{N^{2}({\\tt s}_{\\Gamma_{i}}+2m)^{2}}.\n\\end{displaymath}\n\n\\noindent Integrating we obtain\n\\begin{equation}\\label{ACOM}\n\\frac{dA}{({\\tt s}_{\\Gamma_{i}} +2m)^{2}}\\leq \\frac{dA_{0}}{(2m)^{2}}\\exp (\\int_{0}^{{\\tt s}_{\\Gamma_{i}}}\\frac{2{\\mathcal{M}}_{\\Gamma_{i}}}{N^{2}({\\tt s}_{\\Gamma_{i}}+2m)^{2}}d{\\tt s}_{\\Gamma_{i}}).\n\\end{equation}\n\n\\noindent where $dA_{0}$ is the element of area of $H_{\\Gamma_{i}}$. Recalling that $N^{2}=\\hat{\\st}\/(\\hat{\\st}+2m)$ it is clear that we need an estimation of $\\hat{\\st}$ in terms of ${\\tt s}_{\\Gamma_{i}}$ to have an inequality in terms of ${\\tt s}_{\\Gamma_{i}}$ only. We advocate to that in the following lines. We explain first how to get a relation between ${\\tt s}$ and ${\\tt s}_{\\Gamma_{i}}$ and then we explain how to obtain one in terms of $\\hat{{\\tt s}}$ and ${\\tt s}_{\\Gamma_{i}}$.\n\nFirst recall from (\\ref{C4}) that for any point $q$ in $H_{\\Gamma_{i}}$ we have (for $\\Gamma_{i}$ small enough) that ${\\tt s}(q)=\\hat{\\st}(q)+O(\\hat{\\st}^{\\frac{3}{2}})$. Now let $p$ be a point in $\\gamma$. Then we have\n${\\tt s}(p)\\leq {\\tt s}_{\\Gamma_{i}}(p)+{\\tt s}(q)$, where here $q$ is the initial point of $\\gamma$ at $H_{\\Gamma_{i}}$. Thus ${\\tt s}(p)\\leq {\\tt s}_{\\Gamma}(p)+(1+\\epsilon)\\hat{\\st}(p)$ where $\\epsilon=O(\\hat{\\st}(p)^{\\frac{1}{2}})$. On the other hand let $\\bar{\\gamma}$ be a length minimizing geodesic joining $p$ to $H$. Let $\\bar{q}$ be the point of intersection to $H_{\\Gamma_{i}}$. Then we have \n\\begin{displaymath}\n{\\tt s}(p)=dist(p,\\bar{q})+{\\tt s}(\\bar{q})\\geq {\\tt s}_{\\Gamma_{i}}(p)+\\hat{\\st}(\\bar{q})+O(\\hat{\\st}({\\bar{q}})^{\\frac{3}{2}})\\geq{\\tt s}_{\\Gamma_{i}}(p)+(1-\\epsilon)\\hat{\\st}(q),\n\\end{displaymath}\n\n\\noindent where $\\epsilon=O(\\hat{\\st}(q)^{\\frac{1}{2}})$. Thus for every point $p$ in $\\gamma$ we have\n\\begin{displaymath}\n(1-\\epsilon)\\hat{\\st}_{0}+{\\tt s}_{\\Gamma_{i}}(p)\\leq {\\tt s}(p)\\leq {\\tt s}_{\\Gamma_{i}}(p)+(1+\\epsilon)\\hat{\\st}_{0},\n\\end{displaymath}\n\n\\noindent where we have made $\\hat{\\st}_{0}=\\hat{\\st}(q)$ to simplify the notation. This establishes the relation between ${\\tt s}$ and ${\\tt s}_{\\Gamma_{i}}$. We obtain now the desired relation between ${\\tt s}_{\\Gamma_{i}}$ and $\\hat{\\st}$. We will keep the notation as before. Precisely, $\\gamma$ will be length minimizing geodesic segment to $H_{\\Gamma_{i}}$ and $q$ and $q_{1}$ will be its initial and final points. From Proposition \\ref{GRADUN}, we know that $|\\nabla \\hat{\\st}|\\leq 1$ therefore for any point $p$ between $q$ and $q_{1}$ we have\n\\begin{align*}\n&\\hat{\\st}(q_{1})-\\hat{\\st}(p)\\leq {\\tt s}_{\\Gamma_{i}}(q_{1})-{\\tt s}_{\\Gamma_{i}}(p),\\\\\n&\\hat{\\st}(p)-\\hat{\\st}(q)\\leq {\\tt s}_{\\Gamma_{i}}(p).\n\\end{align*}\n   \n\\noindent Using this we have\n\\begin{displaymath}\n(1+\\epsilon)\\hat{\\st}_{0}\\geq \\hat{\\st}(q)\\geq \\hat{\\st}(p)-{\\tt s}_{\\Gamma_{i}}(p)\\geq \\hat{\\st}(q_{1})-{\\tt s}_{\\Gamma_{i}}(q_{1})\\geq\n\\hat{\\st} (q_{1})-{\\tt s}(q_{1})+(1-\\epsilon)\\hat{\\st}_{0}.\n\\end{displaymath}\n\n\\noindent Now if ${\\tt s}(q_{1})\\geq \\bar{L}$ and $\\bar{L}=\\bar{L}(\\Gamma_{i})$ is big enough we have $\\hat{\\st}(q_{1})-{\\tt s}(q_{1})\\geq -\\epsilon \\hat{\\st}_{0}$. As a result we have the relation\n\\begin{equation}\\label{SSHR}\n(1+\\epsilon)\\hat{\\st}_{0}\\geq \\hat{\\st}(p)-{\\tt s}_{\\Gamma}(p)\\geq (1-2\\epsilon)\\hat{\\st}_{0}.\n\\end{equation}\n\nWe have now all the elements to proceed with the proof of the Proposition. Consider the set of the initial points on $H_{\\Gamma_{i}}$ of the geodesics in ${\\mathcal{F}}_{\\Gamma_{i}}$ whose lengths are greater than $\\bar{L}(\\Gamma_{i})$. Denote such set by $\\Omega_{\\Gamma_{i}}$. We will show now that as $\\Gamma_{i}\\downarrow 0$, and therefore as $H_{\\Gamma_{i}}$ approaches $H$, the area of $\\Omega_{\\Gamma_{i}}$ with respect to the area element induced from $g$ tends to the total area of the horizon $H$.     \n\nConsider the argument in the exponential function of (\\ref{ACOM}) with the upper limit of integration equal to $\\bar{L}$. Using the relation (\\ref{SSHR}) we obtain\n\\begin{align*}\n\\int_{0}^{\\bar{L}}\\frac{{\\mathcal{M}}_{0}}{N^{2}({\\tt s}_{\\Gamma_{i}}+2m)^{2}}d{\\tt s}_{\\Gamma_{i}}&=\\int_{0}^{\\bar{L}} \\frac{{\\mathcal{M}}_{0}(\\hat{\\st}+2m)}{\\hat{\\st}^{2}({\\tt s}_{\\Gamma_{i}}+2m)^{2}}d{\\tt s}_{\\Gamma_{i}}\\\\\n&\\leq\\int_{0}^{\\bar{L}}\\frac{{\\mathcal{M}}_{0}({\\tt s}_{\\Gamma_{i}}+2m+(1+\\epsilon)\\hat{\\st}_{0})}{({\\tt s}_{\\Gamma_{i}}+(1-2\\epsilon)\\hat{\\st}_{0})({\\tt s}_{\\Gamma_{i}}+2m)^{2}}d{\\tt s}_{\\Gamma_{i}}. \n\\end{align*}\n\n\\noindent This last integral can be further split into\n\\begin{displaymath}\n\\int_{0}^{\\bar{L}} \\frac{{\\mathcal{M}}_{0}}{({\\tt s}_{\\Gamma_{i}}+(1-2\\epsilon)\\hat{\\st}_{0})({\\tt s}_{\\Gamma_{i}}+2m)}d{\\tt s}_{\\Gamma_{i}}+R(\\hat{\\st}_{0}),\n\\end{displaymath}\n\n\\noindent where $R(\\hat{\\st}_{0})$ is an expression which is easily seen to tend to zero as $\\hat{\\st}_{0}$ tends to zero. \nWe integrate now equation (\\ref{ACOM}) in $dA$. After integrating in $dA$, the left hand side tends to $4\\pi$ for a suitable divergent sequence of $\\bar{L}$'s. The right hand side is easily integrated to be (discard the term $R(\\hat{\\st}_{0})$)\n\\begin{displaymath}\n\\int_{\\Omega_{\\Gamma_{i}}} \\frac{\\hat{\\st}_{0}}{(\\hat{\\st}_{0}+2m)(2m)^{2}}(\\frac{2m}{(1-2\\epsilon)\\hat{\\st}_{0}})^{\\frac{2{\\mathcal{M}}_{0}}{2m-(1-2\\epsilon)\\hat{\\st}_{0}}}dA_{g},\n\\end{displaymath}\n\n\\noindent where $dA_{g}=N^{2}dA_{0}=\\frac{\\hat{\\st}_{0}}{\\hat{\\st}_{0}+2m}dA$ is the element of area induced on $H_{\\Gamma_{i}}$ from the metric $g$. As a result we get the inequality\n\\begin{equation}\\label{INEF}\n4\\pi \\leq \\frac{\\limsup A(\\Omega_{\\Gamma_{i}})}{4m^{2}} \\limsup \\hat{\\st}_{0}^{\\frac{ 2{\\mathcal{M}}_{0}-2m+(1-2\\epsilon)\\hat{\\st}_{0}}{2m-(1-2\\epsilon)\\hat{\\st}_{0}}}.\n\\end{equation}\n\n\\noindent Now, from the proof of Proposition \\ref{VMH} it is seen that $|{\\mathcal{M}}_{0}-m|\\leq c_{1}\\hat{\\st}^{\\frac{1}{2}}_{0}$ where $c_{1}$ is a positive constant.  Thus we get\n\\begin{displaymath}\n\\hat{\\st}_{0}^{\\frac{ 2{\\mathcal{M}}_{0}-2m+(1-2\\epsilon)\\hat{\\st}_{0}}{2m-(1-2\\epsilon)\\hat{\\st}_{0}}}\\leq \\hat{\\st}_{0}^{c_{2}\\hat{\\st}_{0}^{\\frac{1}{2}}}\\rightarrow 1,\\ {\\rm as}\\ \\hat{\\st}_{0}\\rightarrow 0,\n\\end{displaymath}\n\n\\noindent where $c_{2}$ is a positive constant. Therefore we get from this and equation (\\ref{INEF})\n\\begin{displaymath}\n16\\pi m^{2}\\leq \\limsup A(\\Omega_{\\Gamma_{i}})\\leq A=16\\pi m^{2},\n\\end{displaymath}\n\n\\noindent where $A$ is the area of the horizon. Thus $\\lim\\sup A(\\Omega_{\\Gamma_{i}})=A$. This was the crucial estimate. From it, it will follow that for any $L<\\infty$ fixed, there is a subsequence $\\Gamma_{i_{j}}$ such that the area of the set of initial points in $H_{\\Gamma_{i_{j}}}$ of the geodesics in ${\\mathcal{F}}_{\\Gamma_{i_{j}}}$ whose length is less or equal than $L$, tends actually to zero. This would finish the proof of the Proposition. We do that now. For every $j$, denote by $\\Omega_{L,\\Gamma_{i_{j}}}$ such set. For every $q$ in $\\Omega_{L,\\Gamma_{i_{j}}}$ let $\\gamma_{q}$ be the corresponding geodesic in ${\\mathcal{F}}_{\\Gamma_{i_{j}}}$ whose total length is less than or equal to $L$. Denote by $U_{L,\\Gamma_{i_{j}}}$ the union $U=\\cup_{q\\in\\Omega_{L,\\Gamma_{i_{j}}}} \\{\\gamma_{q}\\}$.  Now, recalling that $dV'=dA$, integrating equation (\\ref{ACOM}), and following the same treatment at the horizon as before gives\n\\begin{displaymath}\nVol_{{\\tt g}}(U_{L,\\Gamma_{i_{j}}})\\leq c(L)A_{g}(\\Omega_{L,\\Gamma_{i_{j}}}).\n\\end{displaymath}\n\n\\noindent Note that in this equation, the volume is found with ${\\tt g}$ while the area is found with $g$. As $A(\\Omega_{i_{j}})\\rightarrow 0$, the Proposition follows.\\hspace{\\stretch{1}}$\\Box$    \n\n\\vs\nThe Proposition before has the following quite important Corollary.\n\\begin{Corollary}\\label{CORFIN} Let $(\\Sigma,{\\tt g},\\ln N)$ be an asymptotically flat static solution with regular an connected horizon. Then\n\\begin{enumerate}\n\\item ${\\tt s}=\\hat{\\st}$ and therefore ${\\tt s}$ is smooth.\n\\item $|\\nabla \\hat{\\st}|^{2}=1$.\n\\item The integral curves of $\\nabla \\hat{\\st}$ are geodesics minimizing the length between any two of its points.\n\\item The set of integral curves of $\\nabla \\hat{\\st}$ form an integrable congruence of geodesics.\n\\end{enumerate}\n\\end{Corollary}\n\n\\noindent {\\it Proof:}\n\n\\vs\n\tLet $p \\in \\Sigma\\setminus H$. Let $\\{\\Gamma_{i}\\}$ such that $\\Gamma_{i}\\downarrow 0$. Following Proposition \\ref{VOLZ} there is a sequence $\\{\\gamma_{i}\\}$ of length minimizing geodesics to $H_{\\Gamma_{i}}$ with initial point $q_{i}$ (at $H_{\\Gamma_{i}}$), $l(\\gamma_{i})\\rightarrow \\infty$ and $\\gamma_{i}(s(p))\\rightarrow p$. Let $p_{i}$ be either the end point of $\\gamma_{i}$ or, if $l(\\gamma_{i})=\\infty$, a point on $\\gamma_{i}$ such that $s(p_{i})\\rightarrow \\infty$. We have \n\\begin{equation}\\label{SSTD}\n\\hat{\\st}(p_{i})-\\hat{\\st}(q_{i})=\\int_{\\bar{s}(q_{i})=0}^{\\bar{s}(p_{i})}<\\nabla\\hat{\\st},\\gamma'>d\\bar{s}=\\bar{s}(p_{i})-\\bar{s}(q_{i})-\\int_{\\bar{s}(q_{i})}^{\\bar{s}(p_{i})}(1-<\\nabla \\hat{\\st},\\gamma'>)d\\bar{s}.\n\\end{equation}\n\n\\noindent where $\\bar{{\\tt s}}$ is the arc-length. But by Proposition \\ref{DELZ} we have $\\lim \\delta(p_{i})={\\tt s}(p_{i})-\\hat{\\st}(p_{i})=0$ and thus we have $\\lim \\bar{{\\tt s}}(p_{i})-\\hat{\\st}(p_{i})=0$ (note that $\\lim |{\\tt s}(p_{i})-\\bar{{\\tt s}}(p_{i})|=0$). By Proposition \\ref{GRADUN} we have\n$(1-<\\nabla \\hat{\\st},\\gamma'>)\\geq 0$, thus from equation (\\ref{SSTD}) we get\n\\begin{displaymath}\n0\\leq \\lim \\int (1-<\\nabla \\hat{\\st},\\gamma'>)d\\bar{{\\tt s}}=0,\n\\end{displaymath}\n\n\\noindent This shows $|\\nabla {\\tt s}|(p)=1$. Moreover we have \n\\begin{displaymath} \n\\hat{\\st}(p)=\\lim \\hat{\\st}(p_{i})-\\hat{\\st}(q_{i})=\\lim\\ \\bar{{\\tt s}}(p_{i})-\\bar{{\\tt s}}(q_{i})-\\int_{\\bar{{\\tt s}}(q_{i})}^{\\bar{{\\tt s}}(p_{i})} (1-<\\nabla \\hat{\\st},\\gamma'>)d\\bar{{\\tt s}}=\\lim \\bar{{\\tt s}}(p_{i})={\\tt s}(p).\n\\end{displaymath}\n\n\\noindent Because $p$ is an arbitrary point we have thus proved {\\it items 1,2} of the Proposition.\n\nTo prove the third {\\it item} we proceed like this. Let $\\gamma$ be an integral curve of $\\nabla \\hat{\\st}$ with initial point $p$ and final point $q$. Suppose that $\\gamma$ does not minimize the distance between $p$ and $q$, namely that there is another curve $\\tilde{\\gamma}$ joining $p$ and $q$ and having smaller length. Then\n\\begin{displaymath}\n{\\tt s}(q)={\\tt s}(p)+({\\tt s}(q)-{\\tt s}(p))={\\tt s}(p)+l(\\gamma)<{\\tt s}(p)+l(\\tilde{\\gamma})\\leq {\\tt s}(q).\n\\end{displaymath}\n\n\\noindent which is a contradiction. \n\n{\\it Item 4} of the Proposition follows directly from the fact that the congruence is orthogonal to the level set of any regular value of ${\\tt s}$. \\hspace{\\stretch{1}}$\\Box$ \n\n\\subsubsection{The uniqueness of the Schwarzschild solutions.}\n\n\\begin{Theorem}\\label{TUSFIN} Let $(\\Sigma,{\\tt g},\\ln N)$ be an asymptotically flat static solution with regular an connected horizon. Then the solutions is a Schwarzschild solution of positive mass. \n\\end{Theorem}\n\n\\noindent {\\it Proof:}\n\n\\vs\nBy Corollary \\ref{CORFIN} the set of integral curves of $\\nabla \\hat{\\st}$ is an integrable congruence of geodesics. Recalling that $|\\nabla \\hat{\\st}|=1$ and $\\Delta\\hat{\\st}=\\theta$, where $\\theta$ is the mean curvature of the congruence. Using these facts in equation (\\ref{SBF} we get that\n\\begin{displaymath}\n{\\mathcal{M}}_{a=2m}=(\\frac{\\theta({\\tt s}+2m)^{2}}{2}-({\\tt s}+2m))N^{2}=m,\n\\end{displaymath}\n\n\\noindent over any geodesic of the congruence. The conclusion that the solution is the Schwarzschild solution follows from Proposition \\ref{COM} and the Remark after it.\\hspace{\\stretch{1}}$\\Box$ \n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\n\nOne of the most interesting applications of photonic\ncrystals (PhCs) is the localization of light to very small\nmode volumes -- below a cubic optical wavelength.  The principal confinement mechanism is Distributed Bragg Reflection (DBR), in contrast to, e.g., microspheres or microdisks, which rely solely on Total Internal Reflection (TIR).  However, since three-dimensional (3D) PhCs (employing DBR confinement in all 3D in space) have not been perfected yet, the mainstream of the PhC research has addressed 2D\nPhCs of finite depth, employing DBR in 2D and TIR in the remaining 1D. Such structures are also more compatible with the present microfabrication and planar integration techniques. These cavities can still have small mode volumes, but the absence of full 3D confinement by DBR makes the problem of the high-quality factor (high-Q) cavity design much more challenging. The main problem is out-of-plane loss by imperfect TIR, which becomes particularly severe in the smallest volume cavities.\n\nOver the past few years, many approaches have been proposed to address this issue, but they all focused on the optimization of a particular cavity geometry and a particular mode supported by it \\cite{ref:SGJohnson2001,ref:JV2001,ref:JV2002,\nNoda2003,ref:Srinivasan02,ref:Korea02a,ref:Lalanne04,ref:Ryu04,ref:Noda05,ref:Geremia02,ref:Noda2005NatureMaterials}.  Some of the proposed cavities seem promising and have already been\nproven useful as components of lasers or add\/drop filters.  However, a general recipe for designing optimized nanocavities is missing, and it is also not known if cavities better than the ones presently known are possible; these are exactly the issues that we will address in this article.  In Section \\ref{sect2}, we will first derive a simple set of\nequations for calculation of cavity $Q$ and mode volume.  In Section \\ref{sect3}\nwe estimate the optimum $k$-space distribution of the cavity mode field.  In Section \\ref{sect5}, we will finally address the question of finding PhC cavities with maximum possible figures of merit for various applications. In this process, we\nstart from the optimum $k$-space distribution of the cavity field; then we derive an approximate analytical relation between the cavity mode and the dielectric constant along a direction of high symmetry, and use it to create a cavity that supports the selected high-$Q$ mode in a single step. Thus, we eliminate the need for trial and error or other parameter search processes, that are typically used in PhC cavity designs. Furthermore, we study the limit of out-of-plane $Q$ factor for a given $V$ of the\ncavity mode with a particular field pattern.\n\n\\section{Simplified relation between $Q$ of a cavity mode and its $k$-space Distribution}\n\\label{sect2}\n\nIn order to simplify PhC cavity optimization, we derive an analytical relation between the near-field pattern of the cavity\nmode and its quality factor in this section.  $Q$ measures how well the cavity\nconfines light and is defined as\n\n\\begin{equation}\n\\label{eq:def_Q} Q \\equiv \\omega \\fr{\\braket{U}}{\\braket{P}}\n\\end{equation}\nwhere $\\omega$ is the angular frequency of the confined mode.  The\nmode energy is\n\\begin{eqnarray}\n\\label{eq:mode_energy} \\braket{U} &=& \\int \\fr{1}{2} (\\varepsilon\nE^{2} + \\mu H^{2}) dV\n\\end{eqnarray}\nThe difficulty lies in calculating $P$, the far-field radiation intensity.\n\nFollowing our prior work \\cite{ref:JV2001}, we consider the\nin-plane and out-of-plane mode loss mechanisms in two-dimensional\nphotonic crystals of finite depth separately: \n\\begin{equation}\n\\braket{P} = \\braket{P_{||}}+\\braket{P_{\\perp}}\n\\end{equation}\nor\n\\begin{equation}\n\\label{eq:Qbreakup}\n\\fr{1}{Q} = \\fr{1}{Q_{||}}+\\fr{1}{Q_{\\perp}}\n\\end{equation}\nIn-plane confinement occurs through DBR.  For frequencies well inside the photonic band gap, this confinement can be made arbitrarily high (i.e., $Q_{||}$ arbitrarily large) by addition of PhC layers.   On the other hand, out-of-plane confinement, which dictates $Q_{\\perp}$, depends on the modal k-distribution that is not confined by TIR.  This distribution is highly sensitive to the exact mode pattern and must be optimized by careful tuning of the PhC defect. Assuming that the cavity mode is well inside the photonic band gap, $Q_{\\perp}$ gives the upper limit for the total $Q$-factor of the cavity mode.\n\nGiven a PhC cavity or waveguide, we can compute the near-field\nusing Finite Difference Time Domain (FDTD) analysis.  As described in Reference \\cite{ref:JV2002}, the near-field pattern at a surface $S$ above the PhC slab contains the complete information about the out-of-plane radiation losses of the mode, and thus about $Q_{\\perp}$ (Fig. \\ref{fig:struct-setup}).  \n\n\\begin{figure}[htbp]\n    \\includegraphics[width=4in]{struct-setup_lowQ.eps}\n \\caption{Estimating the radiated power and $Q_{\\perp}$ from the known near field at\nthe surface $S$} \n\\label{fig:struct-setup}\n\\end{figure}\n\nThe total time-averaged power radiated into the half-space above\nthe surface $S$ is:\n\n\\begin{equation}\nP=\\int\\limits_0^{\\pi\/2} \\int\\limits_{0}^{2\\pi} d\\theta d\\phi \\sin(\\theta) K(\\theta,\\phi) ,\n\\label{eq:prad_general}\n\\end{equation}\n\nwhere $K(\\theta,\\phi)$ is the radiated power per unit solid angle.  In the appendix, we derive a very simple form for $K$ in terms of 2D Fourier Transforms (FTs) of $H_z$ and $E_z$ at the surface $S$, after expressing the angles $\\theta, \\phi$ in terms of $k_x$ and $k_y$:\n\\begin{equation}\n\\label{eq:K_kxky2}\nK(k_x, k_y) = \\fr{\\eta k_z^{2}}{2\n\\lambda^{2}k_{\\|}^{2}} \\left[ \\fr{1}{\\eta^{2}} \\left|\nFT_2(E_z)\\right|^{2}+\n \\left| FT_2(H_z)\\right|^{2} \\right]\n\\end{equation}\n\nHere, $\\eta\\equiv\\sqrt{\\frac{\\mu_o}{\\epsilon_o}}$,  $\\lambda$ is the mode wavelength in air, $k=2\\pi\/\\lambda$, and $\\vec{k}_{||}=(k_x,k_y)=k(\\sin\\theta \\cos\\phi,\\sin\\theta\\sin\\phi)$ and $k_z=k \\cos(\\theta)$ denote the in-plane and out-of-plane $k$-components, respectively.  In Cartesian coordinates, the radiated power (\\ref{eq:prad_general}) can thus be re-written as the integral over the light cone, $k_{||}<k$.  Substituting (\\ref{eq:K_kxky2}) into (\\ref{eq:prad_general}) gives\n\n\\begin{eqnarray}\n\\label{eq:Prad}\nP &\\approx& \\fr{\\eta}{2 \\lambda^{2} k} \\int_{k_{\\|} \\leq k} \\fr{dk_x\ndk_y}{ k_{\\|}^{2}} k_z \\left[ \\fr{1}{\\eta^{2}}  \\left|\nFT_2(E_z)\\right|^{2}+  \\left|FT_2(H_z)\\right|^{2} \\right]\n\\end{eqnarray}\n\nThis is the simplified expression we were seeking.  It gives the out-of-plane radiation loss as the light cone integral of the simple radiation term (\\ref{eq:K_kxky2}), evaluated above the PhC slab.  Substituting Eq. (\\ref{eq:mode_energy}) and Eq. (\\ref{eq:Prad}) into Eq. (\\ref{eq:def_Q}) thus yields a straightforward calculation of the $Q$ for a given mode.  In the following sections, when considering the qualitative behavior of (\\ref{eq:Prad}), we will restrict ourselves to TE-like modes, described at the slab center by the triad $(E_x,E_y,H_z)$, that have $H_z$ even in at least one dimension $x$ or $y$.  For such modes, the term $|FT_2(H_z)|^2$ in (\\ref{eq:Prad}) just above the slab is dominant, and  $|FT_2(E_z)|^2$ can be neglected in predicting the general trend of Q.  \n\nThe figure of merit for cavity design depends on the application: for spontaneous emission rate enhancement through the Purcell effect,\none desires maximal $Q\/V$; for nonlinear optical effects $Q^2\/V$; while for the strong coupling regime of cavity QED, maximizing ratios $g\/\\kappa \\sim Q\/ \\sqrt{V}$ and $g\/\\gamma \\sim 1\/ \\sqrt{V}$ is important. In these expressions, $V$ is the cavity mode volume: $V \\equiv (\\int \\varepsilon(\\vec{r}) |\\vec{E}(\\vec{r})|^{2}\nd^{3}\\vec{r})\/\\max(\\varepsilon(\\vec{r}) |\\vec{E}(\\vec{r})|^{2})$, $g$ is the emitter-cavity field coupling, and $\\kappa$ and $\\gamma$ are the cavity field and emitter dipole decay rates, respectively \\cite{ref:JV2001}.\n\nThus, for a given mode pattern, we have derived a simple set of equations that allow easy evaluations of the relevant figures of merit.  In the next section, we address the problem of finding the field pattern that optimizes these figures of merit and later derive the necessary PhC to support it.  \n\n\n\\section{Optimum $k$-space field distribution of the cavity mode field}\n\\label{sect3}\n\tThe photonic crystals considered here are made by periodically modulating the refractive index of a thin semiconductor slab (see Fig. \\ref{fig:PCslab}). The periodic modulation introduces an energy bandstructure for light in two dimensions $\\vec{r}=(x,y)$. The forbidden energy bands are the source of DBR confinement. \n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[height=1.5in]{TE_sq_hex_BD_BZ_cm_ws_final_lowQ.eps}\n\\caption{\n\\textit{Left:} TE-like mode band diagram for the square lattice photonic crystal. The inset on the right shows the reciprocal lattice (orange circles), the first Brillouin zone (blue) and the irreducible Brillouin zone (green) with the high symmetry points. The inset on the left shows the square lattice PhC slab and relevant parameters: periodicity $a$, hole radius $r$, and slab thickness $d$. The parameters used in the simulation were $r\/a=0.4, d\/a=0.55$, and $n=3.6$.  \n\\textit{Right:} Band diagram for TE-like modes of the hexagonal lattice PhC. The inset on the right shows the reciprocal lattice (orange), the Brillouin zone (blue), and the first irreducible Brillouin zone (green) for the hexagonal lattice photonic crystal. The inset on the left shows the hexagonal lattice PhC slab. The parameters used for the simualtion were $r\/a=0.3, d\/a=0.65, n=3.6$.  A discretization of 20 points per period $a$ was used for both diagrams. }\n\\label{fig:PCslab}\n\\end{center}\n\\end{figure}\n\nThe periodic refractive index\n$\\epsilon(\\vec{r})=\\epsilon(\\vec{r}+\\vec{R})$ can be expanded in a Fourier sum over spatial frequency components in the periodic plane:\n\n\\begin{equation}\\epsilon(\\vec{r})=\\sum_{\\vec{G}}\\epsilon_{\\vec{G}}e^{i\\vec{G}\\cdot{\\vec{r}}}\\end{equation}\n\nHere $\\vec{G}$ are the reciprocal lattice vectors in the\n$(k_{x},k_{y})$ plane and are defined by\n$\\vec{G}\\cdot\\vec{R}=2\\pi{m}$ for integer $m$. The real and\nreciprocal lattice vectors for the square and hexagonal lattices\nwith periodicity $a$ are:\n\n\\indent Square Lattice:\n\\begin{eqnarray}\n\\vec{R}_{mj}=ma\\hat{x}+ja\\hat{y}\\\\\n\\vec{G}_{ql}=\\frac{2\\pi{q}}{a}\\hat{x}+\\frac{2\\pi{l}}{a}\\hat{y}\n\\nonumber\\end{eqnarray} \\indent Hexagonal Lattice:\n\\begin{eqnarray}\n\\vec{R}_{mj}=ma\\frac{(\\hat{x}+\\hat{y}\\sqrt{3})}{2}+ja\\frac{(\\hat{x}-\\hat{y}\\sqrt{3})}{2}\\\\\n\\nonumber\n\\vec{G}_{ql}=\\frac{4\\pi}{a\\sqrt{3}}q\\frac{(-\\hat{x}\\sqrt{3}+\\hat{y})}{2}+\\frac{4\\pi}{a\\sqrt{3}}l\\frac{(\\hat{x}\\sqrt{3}+\\hat{y})}{2},\n\\nonumber\\end{eqnarray}\n\nwhere $m$, $j$, $q$ and $l$ are integers. The electromagnetic\nfield corresponding to a particular wave vector $\\vec{k}$ inside\nsuch a periodic medium can be expressed as \\cite{YarivYeh}:\n\n\\begin{equation}\n\\vec{E}_{\\vec{k}}=e^{i\\vec{k}\\cdot{\\vec{r}}}\\sum_{\\vec{G}}A_{\\vec{k},\\vec{G}}e^{i\\vec{G}\\cdot{\\vec{r}}}\n\\end{equation}\n\\newline\n\nBy introducing linear defects into a PhC lattice, waveguides can\nbe formed. Waveguide modes have the discrete translational\nsymmetry of the particular direction in the PhC lattice and can\ntherefore be expanded in Bloch states $E_{\\vec{k}}$. In Fig.\n\\ref{fig:hex_modes_cav_wg}, we plot the dispersion of a waveguide in the $\\Gamma{J}$ direction of a hexagonal lattice PhC.\n\n\\begin{figure}[htbp]\n\\begin{center}\n \\includegraphics[width=5.0in]{hexWGcav_lowQ.eps}\n   \\caption{\\textit{Left:} Band diagram for hexagonal waveguide in $\\Gamma{J}$ direction, with $r\/a=0.3$, $d\/a=0.65$, $n=3.6$.  The bandgap (wedged between the gray regions) contains three modes.  Mode $B_{oo}$ can be pulled inside the bandgap by additional neighbor hole tuning.\n\\textit{Right:} $B_z$ of confined modes of hexagonal waveguide.  The modes are indexed by the $B$-field's even (``e'') or odd (``o'') parities in the $x$ and $y$ directions, respectively.  The confined cavity modes $B_{oo}$, $B_{ee}$, and $B_{eo}$ required additional structure perturbations for shifting into the bandgap.  This was done by changing the diameters of neighboring holes.  }\n\\end{center}\n\\label{fig:hex_modes_cav_wg}\n\\end{figure}\n\nCavity modes can then be formed by closing a portion of a\nwaveguide, i.e., by introducing mirrors to confine a portion of the waveguide mode. In case of perfect mirrors, the standing wave is described by\n$H=a_{k_0}H_{k_0}+a_{-k_0}H_{-k_0}$. (Here we focus on TE-like PhC\nmodes, and discuss primarily $H_z$, although similar relations can\nbe written for all other field components). Imperfect mirrors\nintroduce a phase shift upon reflection; moreover, the reduction\nof the distance between the mirrors (shortening of the cavity)\nbroadens the distribution of $k$ vectors in the mode to some width\n$\\Delta k$. The $H_z$ field component of the cavity (i.e., a\nclosed waveguide) mode can then be approximated as:\n\\begin{equation}\n\\label{eq:k-space-expansion}\nH_z(x,y)\\sim\\sum_{\\vec{G}}\\sum^{\\vec{k}_{0}+\\Delta{\\vec{k}}\/2}_{\\vec{k}_{0}-\\Delta{\\vec{k}}\/2}\\left(A_{\\vec{k},\\vec{G}}e^{i\\vec{k}\\cdot{\\vec{r}}}+A_{-\\vec{k},\\vec{G}}e^{-i\\vec{k}\\cdot{\\vec{r}}}\\right)e^{i\\vec{G}\\cdot{\\vec{r}}}\n\\end{equation}\n\nA similar expansion of $H_z$ can be made at the surface $S$ directly above the\nPhC slab (see Fig. \\ref{fig:struct-setup}), which is relevant for\ncalculation of radiation losses. The Fourier transform of the\nabove equation gives the $k$-space distribution of the cavity mode,\nwith coefficients $A_{\\vec{k},\\vec{G}}$ and\n$A_{-\\vec{k},\\vec{G}}$. The distribution peaks are\npositioned at $\\pm{\\vec{k_o}}\\pm{\\vec{G}}$, with widths directly\nproportional to $\\Delta k$. The $k$-space distribution is\nmapped to other points in Fourier space by the reciprocal lattice\nvectors $\\vec{G}$. To reduce radiative losses, the mapping of\ncomponents into the light cone should be minimized\n\\cite{ref:JV2002}. Therefore, the center of the mode distribution\n$\\vec{k_{0}}$ should be positioned at the edge of the first\nBrillouin zone, which is the region in $k$-space that cannot be\nmapped into the light cone by any reciprocal lattice vector\n$\\vec{G}$ (see Fig. \\ref{fig:PCslab}). For example, this region\ncorresponds to $\\vec{k_{0}}= \\pm j_x \\frac{\\pi}{a}\\hat{k_x}\\pm j_y\n\\frac{\\pi}{a}\\hat{k_y}$ for the square lattice, where $j_x, j_y\n\\in {0,1}$. Clearly, $|j_x|= |j_y|=1$ is a better choice for\n$\\vec{k_{0}}$, since it defines the edge point of the 1st\nBrillouin zone which is farthest from the light cone. Thus, modes centered at this point and $k$-space broadened due to confinement, will radiate the least. Similarly, the optimum $\\vec{k_0}$ for the cavities resonating in the $\\Gamma J$ direction of the hexagonal lattice is\n$\\vec{k_{0}}= \\pm \\frac{\\pi}{a}\\hat{k_x}$ (as it is for the cavity\nfrom Ref. \\cite{Noda2003}, and for the $\\Gamma X$ direction is\n$\\vec{k_{0}}= \\pm \\frac{2\\pi}{a\\sqrt{3}}\\hat{k_y}$ (as it is for\nthe cavity from Refs. \\cite{ref:JV2001} and \\cite{ref:JV2002}).\n\nAssuming that the optimum choice of $\\vec{k}_0$ at the edge of the\nfirst Brillouin zone has been made, the summation over $\\vec{G}$\ncan be neglected in Eq. (\\ref{eq:k-space-expansion}), because it\nonly gives additional Fourier components which are even further\naway from the light cone and do not contribute to the\ncalculation of the radiation losses. In that case, we can express\nthe field distribution as:\n\n\\begin{equation}\n\\label{eq:k-space-expansion1} H_z(x,y) = \\int \\int dk_x dk_y A(k_x,k_y)\ne^{i\\vec{k}\\cdot{\\vec{r}}},\n\\end{equation}\nwhere $A(k_x,k_y)$ is the Fourier space envelope of the mode,\nwhich is some function centered at $\\vec{k_0}=(\\pm k_{0x},\\pm\nk_{0y})$ with the full-width half-maximum (FWHM) determined by\n$\\Delta\\vec{k}=(\\Delta k_x,\\Delta k_y)$ in the $k_x$ and $k_y$\ndirections respectively. Eq. (\\ref{eq:k-space-expansion1}) implies\nthat $H_z(x,y)$ and $A(k_x,k_y)$ are related by 2D Fourier\ntransforms. For example, if $A(k_x,k_y)$ can be approximated by a\nGaussian centered at $\\vec{k_0}=(k_{0x},k_{0y})$ and with the FWHM\nof $(\\Delta k_x,\\Delta k_y)$, the real space field distribution\n$H_z(x,y)$ is a function periodic in the $x$ and $y$ directions\nwith the spatial frequencies of $k_{0x}$ and $k_{0y}$,\nrespectively, and modulated by Gaussian envelope with the widths\n$\\Delta x\\sim 1\/\\Delta k_{x}$ and $\\Delta y\\sim 1\/\\Delta k_{y}$.\nTherefore, the properties of the Fourier transforms imply that the\nextent of the mode in the Fourier space $\\Delta k $ is inversely\nproportional to the mode extent in real space (i.e., the cavity\nlength), making the problem of $Q$ maximization even more\nchallenging when $V$ needs to be simultaneously minimized.\nThis has already been attempted in the past for a dipole cavity\n\\cite{ref:JV2001,ref:JV2002} and a linear defect \\cite{ Noda2003, ref:Noda05, ref:Noda2005NatureMaterials},\nby using extensive parameters space search. In the following\nsections, we will design high $Q$ cavities by completely eliminating the need for parameter space searches and iterative trial and error approaches. \n\n\tThere are two main applications of Eq. (\\ref{eq:Prad}). First, this formulation of the cavity $Q$ factor allows us to investigate the theoretical limits of this parameter and its relation to the mode volume of the cavity. Second, it allows us to quantify the effect of our perturbation on the optimization of $Q$ using only one or two layers of the computational field and almost negligible computational time compared to standard numerical methods. We applied Eq. (\\ref{eq:Prad}) to cavities obtained from an iterative parameter space search. These cavities were previously studied in \\cite{ref:JV2002} and  \\cite{Noda2003}. The results for $Q$ using Eq. (\\ref{eq:Prad}) at S, as well as full first-principle FDTD simulations are shown in Fig. \\ref{fig:QV_est}. A good match is observed. Therefore, our expression  (\\ref{eq:Prad}) is a valid measure of the radiative properties of the cavity and can be used to theoretically approach the design problem; we can also use this form to speed up the optimization of the cavity parameters. The discrepancy between Eq. (\\ref{eq:Prad}) and FDTD is primarily due to discretization errors. \t\n\n\\begin{figure}[htbp]\n\\begin{center}\n \t\\includegraphics[width=2.0in]{Qest_u660_cav_cm_one_lowQ.eps}\n\t\\includegraphics[width=2.0in]{Qest_uDIP_cav_cm_one_lowQ.eps}\n\n \n  \n   \n \n  \n \n \n \n \n\\end {center}\n    \\caption{Comparison of $Q$ factors derived from Eq. (\\ref{eq:Prad}) (squares) to those calculated with FDTD (circles). \\textit{Left:} cavity made by removing three holes along the $\\Gamma{J}$ direction confining the $B_{oe}$ mode. The $Q$ factor is tuned by shifting the holes closest to the defect as shown by the red arrow.  The $x$-axis gives the shift as a fraction of the periodicity $a$. \\textit{Right:} the $X$ dipole cavity described in \\cite{ref:JV2002}. The $Q$ factor is tuned by stretching the center line of holes in the $\\Gamma{X}$ direction, as shown by the arrow. The $x$-axis gives the dislocation in terms of the periodicity $a$. }\n    \\label{fig:QV_est}\n\n\\end{figure}\n\n\\section{Inverse problem approach to designing PhC cavities}\n\\label{sect5}\n\nIn the inverse approach, we begin with a desired in-plane Fourier decomposition of the resonant mode, $FT_2(\\vec{H}(\\vec{r}))$, chosen again to minimize radiation losses given by Eq. (\\ref{eq:Prad}).  The difficulty lies with designing a structure that supports the field which is approximately equal to the target field, $\\vec{H}(\\vec{r})$.  \n\nIn this section, we first estimate the general behavior of $Q\/V$ for structures of varying mode volume.  Then we present two approaches for analytically estimating the PhC structure $\\epsilon(\\vec{r})$ from the desired $k$-space distribution $FT_2(\\vec{H}(\\vec{r}))$.  As mentioned in Sec.\\ref{sect2}, we restrict the analysis to TE-like modes $B_{eo}, B_{oe}$, and $B_{ee}$ (Fig.\\ref{fig:hex_modes_cav_wg}(b)) for which we can approximate the trend of the radiation (\\ref{eq:Prad}) by considering only $H_z$ at the surface $S$ just above the PhC slab.  Moreover, to make a rough estimate of the cavity dielectric constant distribution from the desired $H_z$ field on $S$, we approximate that $H_z$ at $S$ is close to $H_z$ at the slab center.\n\\subsection{General trend of $Q\/V$}\n\nThe simplification described above allows us to study the general behavior of $Q\/V$ for a cavity with varying mode volume.  Here, we assume that a structure has been found to support the desired field $H_z$.  \n\nWe again start from the expression for radiated power, Eq. (\\ref{eq:Prad}), and calculate$Q$ using Eq. (\\ref{eq:def_Q}).  All that is required of the cavity field is that its FT at the surface S above the slab be distributed around the four points $k_{x0}=\\pm \\pi\/a, k_{y0}=\\pm \\fr{2\\pi}{\\sqrt{3}a}$, to minimize the components inside the light cone.  As an example, we choose a field with a Gaussian envelope in Fourier space, as described in Sec. \\ref{sect3}.  For now, let us consider mode symmetry $B_{oe}$. The Fourier Transform of the $H_z$ field is then given by \n\\begin{equation}\n\\label{eq:Gaussian_field}\nFT_2(H_z)=\\sum_{k_{x0},k_{y0}} sign(k_{x0}) \\exp(-(k_x-k_{x0})(\\sigma_x\/\\sqrt{2})^{2}-(k_y-k_{y0})(\\sigma_y\/\\sqrt{2})^{2}),\n\\end{equation}\nwhere $\\sigma_x$ and $\\sigma_y$ denote the modal width in real space.  The mode and its FT are shown in Fig.\\ref{fig:Ba1_calc} (d-e).  We use Eq.\\ref{eq:Prad} without the $E_z$ terms to estimate the trend in $Q$, as described above.  As the mode volume grows, the radiation inside the light cone shrinks exponentially.  This results in an exponential increase in $Q$.  This relationship is shown in Fig.\\ref{fig:Ba1_calc}(g) for field $B_{oe}$ at frequency $a\/\\lambda=0.248$.  At the same time, the mode volume grows linearly with $\\sigma_x$.  The growth of $Q\/V$ is therefore dominantly exponental     , and we can find the optimal $Q$ for a particular choice of mode volume (i.e. $\\sigma_x$) of the Gaussian mode cavity.\n\n\n\\begin{figure}[htbp]\n\n\\includegraphics[width=5.5in]{FieldPlotsX_lowQ.eps}\n   \\caption{Idealized cavity modes at the surface S above the PhC slab; all with mode volume $\\sim (\\lambda\/n)^{3}$. (a-c) Mode with sinc and Gaussian envelopes in $x$ and $y$, respectively: $H_z(x,y)$, $FT_2(H_z)$, and $K(k_x,k_y)$ inside the light cone; (d-g) Mode $B_{oe}$ with Gaussian envelopes in the x and y directions : $H_z(x,y)$, $FT_2(H_z)$, $K(k_x,k_y)$, and $Q_{\\perp}( \\sigma_{x}\/a) $ as well as $Q_{\\perp}\/V$ . $Q_{\\perp}$ was calculated using Eq. (\\ref{eq:Prad}) and $E_{z}$ was neglected.; (h-i) Mode $B_{ee}$ with Gaussian envelopes in x and y can be confined to radiate preferentially upward. } \n\n\n\\label{fig:Ba1_calc}\n\n\n\\end{figure}\n\n\nAccording to Fig.\\ref{fig:Ba1_calc}(g), very large $Qs$ can be reached with large mode volumes and there does not seem to be an upper bound on $Q_{\\perp}$. As the mode volume of the Gaussian cavity increases, the radiative Fourier components vanish exponentially, but are never zero. A complete lack of Fourier components in the light cone should result in the highest possible Q. As an example of such a field, we propose a mode with a sinc envelope in $x$ and a Gaussian one in $y$. The FT of this mode in Fig.\\ref{fig:Ba1_calc}(b) is described by \n \n\\begin{equation}\n\\label{eq:sinccos_field}\nFT_2(H_z)=\\sum_{k_{x0},k_{y0}} \\exp(-(k_y-k_{y0})^{2}(\\sigma_y\/\\sqrt{2})^{2}) Rect(k_x-k_{x0},\\Delta k_x),\n\\end{equation}\nwhere $Rect(k_x,\\Delta k_x)$ is a rectangular function of width $\\Delta k_x$ and centered at $k_x$.\n\nThe Fourier-transform implies that the cavity mode is described by a sinc function in $x$ whose width is inversely proportional to the width of the rectangle $Rect(k_x,\\Delta k_x)$.   To our knowledge, this target field has not been previously considered in PhC cavity design. This field is shown in Fig.\\ref{fig:Ba1_calc}(a-c).  Though it has no out-of-plane losses, this field drops off as $\\fr{1}{r}$ and therefore requires a larger structure than the Gaussian field for confinement.  \n\nOver the past years, many new designs with ever-higher theoretical quality factors have been suggested \\cite{ref:Noda2005NatureMaterials}.  In light of our result that $Q\/V$ increases exponentially with mode size, these large $Q$s are not surprising.  On the other hand, direct measurements on fabricated PhC cavities indicate that $Q$s are bounded to currently $\\sim 10^{4}$ by material absorption and surface roughness \\cite{imamoglu2005,englund2005}, while indirect measurements using waveguide coupling indicate values up to $6\\cdot 10^{5}$\\cite{ref:Noda05, ref:Noda2005NatureMaterials}.  \n\nIt is interesting to note that Eq. (\\ref{eq:K_kxky2}) also allows one to calculate the field required to radiate with a desired radiation distribution.  For example, many applications require radiation with a strong vertical component; waveguide modes with even $H_z$ can be confined for this purpose so that $K(k_x,k_y)$ dominates losses at the origin in $k-$space, as shown for instance in Fig. \\ref{fig:Ba1_calc}(h-i) for the confined mode pattern $B_{ee}$ of Fig.\\ref{fig:hex_modes_cav_wg}.\n\n\\subsection{Estimating Photonic Crystal Design from $k$-space Field Distribution}\n\nNow we introduce two analytical ways of estimating the dielectric structure $\\epsilon(\\vec{r})$ that supports a cavity field that is approximately equal to the desired field $\\vec{H}_c$. These methods directly calculate the dielectric profile from the desired field distribution, without any dynamic tuning of PhC parameters, and are thus computationally fast. We focus on TE-like modes, since they see a large bandgap and exhibit electric-field maxima at the slab center. For TE-like modes, $\\vec{H}_c=H_c\\hat{z}$ at the center of the slab, and $\\vec{H}_c \\approx H_c\\hat{z}$ at the surface. First, we relate $\\vec{H}_c$ to one of the allowed waveguide fields $\\vec{H}_w$.  The fields $H_c$ and $H_w$ at  the center of the PhC slab $(z=0)$ are solutions to the homogeneous wave equation with the corresponding refractive indices $\\epsilon_c$ and $\\epsilon_w$, respectively.\n\n\\begin{equation}\n\\label{eq:Hc}\n- \\mu_0 \\frac{\\partial^2 \\vec{H_c}}{\\partial{t}^2} =\\omega^{2}_c \\mu_{0} \\vec{H}_{c} = \\nabla \\times \\frac{1}{\\epsilon_c} \\nabla \\times \\vec{H}_c \\\\\n\\end{equation}      \n\n\\begin{equation}\n\\label{eq:Hw}\n- \\mu_0 \\frac{\\partial^2 \\vec{H_w}}{\\partial{t}^2} =\\omega^{2}_w \\mu_{0} \\vec{H}_{w} = \\nabla \\times \\frac{1}{\\epsilon_w} \\nabla \\times \\vec{H}_w\n\\end{equation}      \n\n\n\nHere $\\omega_c$ and $\\omega_w$ are the frequencies of the cavity and waveguide fields. We expand the cavity mode into waveguide Bloch modes:\n\\begin{equation} \n\\label{eq:cav_mode}\nH_c=\\sum_{\\vec{k}}c_{k}u_{k}(\\vec{r})e^{i(\\vec{k}\\cdot\\vec{r}-\\omega_{k}t)} \n\\end{equation}\nwhere $u_{k}$ is the periodic part of the Bloch wave. Assuming that the cavity field is composed of the waveguide modes with $\\vec{k} \\approx \\vec{k}_0$, we can approximate $u_k(\\vec{r}) \\approx u_{k0}(\\vec{r})$, which leads to the slowly varying envelope approximation:\n\\begin{equation}\nH_c \\approx u_{k_0}(\\vec{r}) e^{i(\\vec{k_0}\\cdot\\vec{r}-\\omega_{w} t)}\\sum_{\\vec{k}} c_{k} e^{i((\\vec{k}-\\vec{k_0})\\cdot\\vec{r}-(\\omega_{k}-\\omega_{w})t)} =  H_{w} H_{e} ,\n\\label{eq:cav_mode_expand}\n\\end{equation}\nwhere the waveguide mode $H_{w}=u_{k_0}(\\vec{r}) e^{i (\\vec{k_0} \\cdot \\vec{r}-\\omega_{w} t) } $ and the cavity field envelope $H_{e} = \\sum_{\\vec{k}} c_{k} e^{i((\\vec{k}-\\vec{k_0})\\cdot\\vec{r} - (\\omega_{k}-\\omega_{w} ) t )}$. \n\nThe cavity and waveguide fields FT-distributions are concentrated at the edge of the Brillouin zone, where $\\omega^{2}_{k} \\approx \\omega^{2}_{w}+\\alpha\\left|\\vec{k}-\\vec{k}_{0}\\right|^2$ and $\\alpha \\ll 1$ (i.e., the band is nearly flat).  Differentiating (\\ref{eq:cav_mode_expand}) in time twice gives: \n\\begin{eqnarray}\n\\frac{\\partial^2 H_c}{\\partial{t}^2} = -\\omega^{2}_{c} H_c = -\\sum_{\\vec{k}}c_{k}u_{k}(\\vec{r})e^{i(\\vec{k}\\cdot\\vec{r}-\\omega_{k}t)} \\left[\\omega^{2}_{w}+\\alpha\\left|k-k_{0}\\right|^2\\right]  \\approx \\nonumber \\\\\n- H_w \\sum_{\\vec{k}} { c_{k} e^{i((\\vec{k}-\\vec{k_0})\\cdot\\vec{r} - (\\omega_{k}-\\omega_{w}) t )} \\left[\\omega^{2}_{w}+\\alpha\\left|k-k_{0}\\right|^2\\right] } = -\\omega^{2}_{w} H_w H_e+\\alpha H_w \\nabla^2 H_e\n\\label{eq:diff_cav_mode1}\n\\end{eqnarray}\n\nThus, for $\\alpha\\ll 1$ and finite $ \\nabla^2 H_e $, $\\omega_c \\approx \\omega_w$, i.e., the cavity resonance is very close to the frequency of the dominant waveguide mode. The condition $\\alpha\\ll 1$, also implies that $\\omega_k \\approx \\omega_w$, i.e. $H_{e} \\approx \\sum_{\\vec{k}} c_{k} e^{i(\\vec{k}-\\vec{k_0})\\cdot\\vec{r}}$.\n\\subsubsection{Estimating Photonic Crystal Design from $k$-space field Distribution: Approach 1}\n\nLet us express the cavity dielectric constant $\\epsilon_c$ as $\\epsilon_c = \\epsilon_w\\epsilon_e$, where $\\epsilon_e$  is the unknown envelope. $H_c$ is a solution of Eq. (\\ref{eq:Hc}) and each waveguide mode $u_k(\\vec{r})e^{i( \\vec{k} \\cdot \\vec{r}-\\omega_k t)}$ satisfies Eq. (\\ref{eq:Hw}). From previous arguments, for $\\vec{k}$ within the range corresponding to the cavity mode, $\\omega_c \\approx \\omega_w \\approx \\omega_{k}$. Thus, a linear superposition of waveguide modes $\\sum_{k}c_k u_k(\\vec{r})e^{i( \\vec{k} \\cdot \\vec{r}-\\omega_k t)}=H_e H_w=H_c$ also satisfies Eq. (\\ref{eq:Hw}), i.e. the cavity mode is also a solution of Eq. (\\ref{eq:Hw}) for a slowly varying envelope. \n\tWe assume that the mode is TE-like, so that the $H$ field only has a $z$ component at the center of the slab. Then inserting $H_c$ from (\\ref{eq:cav_mode_expand}) into Eq.(\\ref{eq:Hc}) and Eq. (\\ref{eq:Hw}) and subtracting the two equations with $\\epsilon_c = \\epsilon_w\\epsilon_e$, yields a partial differential equation for $\\epsilon_c(x,y)$:  \n\\begin{equation}\n\\label{eq:eps_prod_eq}\n\\partial_x \\biggl[\\fr{1}{\\epsilon_w}\\biggl( \\fr{1}{\\epsilon_e}-1\\biggr) \\partial_x H_c\\biggr] +\\partial_y\\biggl[\\fr{1}{\\epsilon_w}\\biggl( \\fr{1}{\\epsilon_e}-1\\biggr) \\partial_y H_c\\biggr] \\approx \\mu_0 (\\omega^{2}_w-\\omega^{2}_c) H_c \\approx 0\n\\end{equation}\t\n\t\n\tFor this approach, we consider waveguide modes with $B_{oe}$ symmetry in Fig.\\ref{fig:hex_modes_cav_wg}(b).  These modes are even in $\\hat{y}$, so the partial derivatives in $y$ in (\\ref{eq:eps_prod_eq}) vanish at $y=0$.  The resulting simplified first-order differential equation in $1\/\\epsilon_e$ can then be solved directly near $y=0$, and the solution is $\\fr{1}{\\epsilon_w}( \\fr{1}{\\epsilon_e}-1) \\partial_x H_c \\approx C$\t, where $C$ is an arbitrary constant of integration. In our analysis $\\epsilon_w$ corresponds to removing holes along one line ($x$-axis) in the PhC lattice. The cavity is created by introducing holes into this waveguide, which means that $\\frac{1}{\\epsilon_e}-1 > 0$. The solution holds when we take the absolute value of both its sides, and for $C > 0$, this leads to the following result for the cavity dielectric constant near $y=0$:\n\\begin{equation}\n\\label{eq:eps_approach1}\n\\epsilon_c(x) \\approx \\frac{\\left|\\partial_{x}H_c\\right|}{C+\\frac{1}{\\epsilon_{w}}\\left|\\partial_{x}H_c\\right|}\n\\end{equation}\n$C$ is a positive constant of integration, and $H_c=H_w H_e$, where $H_w$ is the known waveguide field and $H_e$ is the desired field envelope. $C$ can be chosen by fixing the value of $\\epsilon_c$ at some x, leading to a particular solution for $\\epsilon_c$. In our cavity designs we chose $C$ such that the value of $\\epsilon_c$ is close to $\\epsilon_w$ at the cavity center. To implement this design in a practical structure, we need to approximate this continuous $\\epsilon_c$ by means of a binary function with low and high-index materials $\\epsilon_{l}$ and  $\\epsilon_{h}$, respectively.  We do this by finding, in every period $j$, the air hole radius $r_j$ that gives the same field-weighted averaged index on the $x$-axis: \n\\begin{equation}\n\\int_{j a -a\/2}^{j a + a\/2} (\\epsilon_h+(\\epsilon_l-\\epsilon_h)Rect(j a,r_j))\\left|\\vec{E}_c\\right|^2 dx = \\int_{j a -a\/2}^{j a + a\/2} \\epsilon_c(x)\\left|\\vec{E}_c\\right|^2 dx ,\n\\end{equation}\nwhere $\\vec{E}_c$ is estimated from a linear superposition of waveguide modes as $\\vec{E}_c\\propto \\nabla\\times \\vec{H}_c$. We assume that the holes are centered at the positions of the unperturbed hexagonal lattice PhC holes. \n\nThe radii $r_j$ thus give the required index profile along the $x$ symmetry axis.  The exact shape of the holes in 3D is secondary -- we choose cylindrical holes for convenience.  Furthermore, we are free to preserve the original hexagonal crystal structure of the PhC far away from the cavity where the field is vanishing.  \n\n\nTo illustrate the power of this inverse approach, we now design PhC cavities that support the Gaussian and sinc-type modes of Eq. (\\ref{eq:Gaussian_field}), (\\ref{eq:sinccos_field}).  In each case, we start with the waveguide field $B_{oe}$ of Fig.\\ref{fig:hex_modes_cav_wg}(b) confined in a line-defect of a hexagonal PhC.  The calculated dielectric structures and FDTD simulated fields inside them are shown in Fig. \\ref{fig:approach1}.  The FT fields on S also show a close match and very little power radiated inside the light cone (Fig.\\ref{fig:approach1}(c,f)).  This results in very large $Q$ values, estimated from $Q_{\\perp}$ to limit computational constraints.  These estimates were done in two ways, using first principles FDTD simulations \\cite{ref:JV2001}, and direct integration of lossy components by Eq. (\\ref{eq:Prad}).  The results are listed in Table \\ref{table:Q_derived} and show an improvement of roughly three orders of magnitude over the unmodified structure of Fig. \\ref{fig:QV_est} with as small increase in mode volume. Furthermore, a fit of the resulting field pattern to a Gaussian envelope multiplied by a Sine, yielded a value of $\\sigma_x\/a \\approx 1.6$, which, according to the plot in Fig. \\ref{fig:Ba1_calc} g., puts us at the attainable limit of $Q_{\\perp}$ at this mode volume.\n\nIn our FDTD simulations, we verified that $Q_{\\perp}$ correctly\nestimates $Q$ by noting that $Q_{\\perp}$ did not change appreciably\nas the number of PhC periods in the $x-$ and $y-$ directions, $N_x$\nand $N_y$, was increased:  for the Gaussian-type (sinc-type) mode,\nincreasing the simulation size from $N_x=13,N_y=13$ ($N_x=21,N_y=9$) PhC\nperiods to $N_x=25,N_y=13$ ($N_x=33,N_y=13$) changed quality factors from\n$Q_{||}=22\\cdot 10^3,Q_{\\perp}=1.4\\cdot 10^6$ ($Q_{||}=17\\cdot 10^3,Q_\n{\\perp}=4.2\\cdot 10^6$) to $Q_{||}=180\\cdot 10^3,Q_{\\perp}=1.48\\cdot\n10^6$ ($Q_{||}=260\\cdot 10^3,Q_{\\perp}=4.0\\cdot 10^6$).  (The number of\nPhC periods in the x-direction in which the holes are modulated to\nintroduce a cavity is 9 and 29 for Gaussian and\nsinc cavity, respectively, while both cavities consist of only one line\nof defect holes in the y-direction.) Thus, with enough periods, the\nquality factors would be limited to $Q_{\\perp}$, as summarized in the\ntable. In the calculation of $Q$, the vertically emitted power\n$\\braket{P_{||}}$ was estimated from the fields a distance $\\sim\n0.25\\cdot \\lambda$ above the PhC surface.  Note that the frequencies $a\/\\lambda$ closely match those of the original waveguide field $B_{oe}$ ($a\/\\lambda_{cav}$=0.251), validating the assumption in the derivation.  \n\n\\begin{table}[htdp]\n\\caption{Q values of structures derived with inverse-approach 1}\n\\begin{center}\n\\begin{tabular}{c|c|c|c|c}\n  & $a\/\\lambda_{cav}$  & $Q_{cav}$ (freq. filter) & $Q_{cav}$ (Eq. (\\ref{eq:Prad})) & $V_{mode} (\\frac{\\lambda}{n})^3  $ \\\\\n \\hline\nGaussian & 0.248 & $1.4\\cdot 10^{6}$ & $1.6\\cdot 10^{6}$ & $0.85$ \\\\\nSinc & 0.247 & $4.2\\cdot 10^{6} $& $4.3 \\cdot 10^{6} $  & $1.43$ \\\\\nUnmodified 3-hole defect & 0.251 & $6.6\\cdot 10^{3} $& $6.4\\cdot 10^{3}$   & $0.63$ \\\\\n\\end{tabular}\n\\end{center}\n\\label{table:Q_derived}\n\\end{table}\n\n\n\\begin{figure}[htbp]\n\\renewcommand{\\baselinestretch}{1.0}\n  \\begin{center}\n \n \\includegraphics[width=5in]{SINC_GAUSS_FIELDS_FITS_lowQ.eps}\n \n\n \n  \n  \n   \n     \n    \n     \n     \n    \\caption{ FDTD simulations for the derived Gaussian cavity (a-c) and the derived sinc cavity (d-f).  Gaussian: (a) $B_z$; (b) $|E|$; (c) FT pattern of $B_z$ taken above the PhC slab (blue) and target pattern (red).  Sinc: (d) $B_z$; (e) $|E|$; (f) FT pattern of $B_z$ taken above the PhC slab (blue) and target pattern (red) (The target FT for the sinc cavity appears jagged due to sampling, since the function was expressed with the resolution of the simulations). The cavities were simulated with a discretization of 20 points per period a, PhC slab hole radius $r = 0.3 a$, slab thickness of $0.6 a$ and refractive index $3.6$. Starting at the center, the defect hole radii in units of periodicity a are: $(0, 0, 0.025, 0.05, 0.075, 0.1, 0.075, 0.075, 0.1, 0.125, 0.125, 0.125, 0.1, 0.125, 0.15, 0.3, 0.3)$ for the sinc cavity, and $(0.025, 0.025, 0.05, 0.1, 0.225)$ for the Gaussian cavity.}\n    \\label{fig:approach1}\n  \\end{center}\n  \\renewcommand{\\baselinestretch}{2.0}\n\\end{figure}\n\n\\subsubsection{Estimating Photonic Crystal Design from $k$-space Field Distribution: Approach 2}\n\n\nWe now derive a closed-form expression for $\\epsilon_c(x,y)$ that is valid in the whole PhC plane (instead of the center line only).  Again, begin with the cavity field $\\vec{H}(\\vec{r})=\\hat{z}H_c$ consisting of the product of the waveguide field and a slowly varying envelope, $H_c=H_{w} H_{e},$ and treat the cavity dielectric constant as: $\\fr{1}{\\epsilon_c}=\\fr{1}{\\epsilon_{pert}}+\\fr{1}{\\epsilon_{w}}$.  In the PhC plane, Eq. (\\ref{eq:Hc}, \\ref{eq:Hw}) for a TE-like mode can be rewritten as \n\\begin{eqnarray}\n\\label{eq:HcHw}\n- \\omega^{2}_c \\mu_{0} H_{c} &=& \\nabla \\cdot (\\frac{1}{\\epsilon_c}\\nabla H_c) \\\\\n- \\omega^{2}_w \\mu_{0} H_{w} &=& \\nabla \\cdot (\\frac{1}{\\epsilon_w}\\nabla H_w) \n\\end{eqnarray}      \nMultiplying the last equation by $H_e$, subtracting from the first, and recalling that $\\omega_c\\sim \\omega_w$ yields\n\\begin{eqnarray}\n\\omega^{2}_w \\mu_{0} H_e H_{w} -\\omega^{2}_c \\mu_{0}H_{c}  &=&\\mu_{0}H_{c} (\\omega_w^{2}-\\omega_c^{2}) \\approx 0\\\\ \\nonumber\n&=&\\nabla \\cdot (\\frac{1}{\\epsilon_c} \\nabla H_c)-H_e  \\nabla \\cdot (\\frac{1}{\\epsilon_w} \\nabla H_w)  \\\\\n&\\approx&\\nabla \\cdot (\\fr{1}{\\epsilon_{pert}} \\nabla H_c)\n\\end{eqnarray}\nwhere the last line results after some algebra and dropping spatial derivatives of the slowly varying envelope $H_e$.  This relation is a quasilinear partial differential equation in $1\/\\epsilon_{pert}$.  With boundary conditions that can be estimated from the original waveguide field, this equation can in principle be solved for $\\epsilon_c $ (e.g., \\cite{haberman1987}).  \n\nAlternatively, one can find a formal solution for $\\epsilon_{pert}$ by assuming a vector function $\\vec{\\xi}(\\vec{r})$ chosen to satisfy the boundary conditions, so that \n\n\\begin{equation}\n \\fr{1}{\\epsilon_{pert}} \\nabla H_c = \\nabla\\times \\vec{\\xi}\n \\end{equation}\n or\n\\begin{equation}\n \\fr{1}{\\epsilon_{pert}} = \\fr{\\nabla\\times \\vec{\\xi} \\cdot \\nabla H_c^{*}}{|\\nabla H_c|^{2}}\n \\end{equation}\n This gives is the formal solution of the full dielectric constant $\\epsilon_c=(\\epsilon_{pert}^{-1}+\\epsilon_{w}^{-1})^{-1}$ in the plane of the photonic crystal.  \n\n\\section{Conclusions}\nWe have described a simple recipe for designing two-dimensional photonic crystal cavities.  Although the approach is general, we have demonstrated its utility on the design of cavities with very large $Q> 10^{6}$ and near-minimal mode volume $\\sim (\\lambda\/n)^{3}$. These values follow our theoretically estimated value of $Q_{\\perp}\/V$ for the cavity with the Gaussian field envelope, which means that we were able to find the maximal Q for the given mode volume V under our assumptions. Our approach is analytical, and the results are obtained within a single computational step. We first derive a simple expression of the modal out-of-plane radiative loss and demonstrate its utility by the straightforward calculation of $Q$ factors on several cavity designs.  Based on this radiation expression, the recipe begins with choosing the FT mode pattern that gives the desired radiation losses.  For high-$Q$ cavities with minimal radiative loss inside the light cone, we show that the transform of the mode should be centered at the extremes of the Brillouin Zone, as far removed from the light cone as possible.  Next we proved that for a cavity mode with a Gaussian envelope, $Q\/V$ grows exponentially with mode volume $V$, while the cavity with the sinc field envelope should lead to even higher Q's by completely eliminating the Fourier components in the light cone. Finally, we derived approximate solutions to the inverse problem of designing a cavity that supports a desired cavity mode.  This approach yields very simple design guides that lead to very large $Q\/V$.  Since it eliminates the need for lengthy trial-and-error optimization, our recipe enables rapid and efficient design of a wide range of PhC cavities.   \n\n\\subsection{Acknowledgements}\nThis work has been supported by the MURI Center for photonic quantum\ninformation systems (ARO\/ARDA Program DAAD19-03-1-0199). Dirk Englund\nhas also been supported by the NDSEG fellowship, and Ilya Fushman by the\nNIH training grant\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{\\label{Intro} Introduction}\n\nLattice QCD has made enormous progress over the last years due to computational and algorithmic advances \\cite{Schaefer:2012tq}.\nThis has led to significantly improved lattice calculations of many low-energy observables. \nPresent-day unquenched lattice calculations are performed with quark masses close to or at their physical value \\cite{Abdel-Rehim:2014nka,Bazavov:2014wgs,Durr:2013goa,Aoki:2009ix}. Uncertainties associated with the chiral extrapolation are essentially eliminated in these simulations. \n\nHowever, dynamical lattice QCD with small quark masses may face new problems. One feature of unquenched lattice simulations is the presence of multi-particle states in the correlation functions measured to obtain observables. With the up and down quark masses getting closer to their physical values one expects multi-particle states with additional pions to become a significant excited-state contamination in many correlation functions.  As a simple example consider the two-point function $C(t)$ of a nucleon interpolating field, as it is measured to extract the nucleon mass $M_N$. \nFrom the spectral decomposition the two-point function in a finite spatial volume, projected to zero momentum, is a sum of exponentials,\n\\begin{equation}\\label{ExpAnsatz}\nC(t) = b_0 e^{-M_0 t} + b_1 e^{-M_1 t} + \\ldots\\,.\n\\end{equation}\nThe first exponential provides the exponential decay with the nucleon mass, $M_0=M_N$. All the other terms stem from states with the same quantum numbers as the nucleon, either genuine single-particle excited states or multi-particle states. For sufficiently small pion masses  one expects a nucleon-pion state to be the state with lowest total energy next to the ground state, $M_1\\approx E_N + E_{\\pi}$. For symmetry reasons the nucleon and the pion cannot be at rest. Both have non-zero but opposite spatial momenta  determined by the spatial volume and the boundary conditions imposed in the spatial directions. Still, for sufficiently large spatial volumes the exponent $M_1$ can be smaller than the first  one-particle excited state, associated with the Roper resonance $N^*(1440)$ in infinite volume. Moreover, near physical quark masses the three-particle state containing the nucleon and two pions at rest will have a smaller energy than the one-particle excited state.\n\nThe way to deal with multi-particle states in spectroscopy calculations is well-known. The well-established variational method \\cite{Luscher:1990ck} can be used provided interpolating fields for the multi-particle states are taken into account.\\footnote{Very recent studies that include two-particle nucleon-pion states in the analysis are reported in Refs.\\ \\cite{Verduci:2014csa,Kamleh:2014nxa,Kiratidis:2015vpa}, for example.} \nStill, the more states one takes into account  the larger is the generalized eigenvalue problem one has to solve numerically, and the error bounds for the energies derived in \\cite{Blossier:2009kd} get worse the denser the spectrum is.\n\nIn this paper we provide some analytical results  concerning the nucleon two-point function. As has been pointed out in  Ref.\\ \\cite{Bar:2012ce}, chiral perturbation theory (ChPT) can be employed to obtain an estimate for the ratio $b_1\/b_0$.\\footnote{To our knowledge the idea for using ChPT to study the two-particle-state contributions to nucleon correlation functions was put forward first in Ref.\\ \\cite{Tiburzi:2009zp}.} \nMoreover, to leading order in the chiral expansion one expects the ratio $b_1\/b_0$ to be independent of the a priori unknown low-energy constant (LEC) associated with the particular choice for the interpolating field. In that sense LO ChPT makes a rather definite prediction for $b_1\/b_0$.  \nEven if this result will receive substantial higher order corrections we do obtain a reliable first estimate for the impact of the nucleon-pion-state contribution to the two-point function.\n\nThe results we find for $b_1\/b_0$ are small. For example, for a pion mass satisfying $M_{\\pi} L \\approx 4$ and $M_{\\pi}\/M_N\\approx 0.2$ we find $b_1\/b_0\\approx 0.1$.  Taking into account the additional exponential suppression the two-particle-state contribution in \\pref{ExpAnsatz} contributes at the few-percent level for euclidean times of about 0.5 fm. For larger $t$ and in the effective mass the contribution is even smaller. Whether it is noticeable in practice is then a question of the size of the statistical errors in the lattice data.\n\nThe nucleon-pion-state contribution to the two-point function has been independently computed in Ref.\\ \\cite{Tiburzi:2015tta}. The computation in that reference is performed in heavy baryon chiral perturbation theory (HBChPT), while here we employ the covariant formulation. In particular the chiral expressions for the interpolating fields differ in these two formulations, thus the final results are not the same. However, performing the appropriate expansion of our results we reproduce  the results given in Ref.\\ \\cite{Tiburzi:2015tta}. \n\n\\section{\\label{secQCD} Nucleon two-point correlators in QCD}\n\n\\subsection{General considerations}\\label{ssect:general}\n\nThroughout this article we consider QCD in a finite spatial box.  $L$ denotes the box length in  each direction and periodic boundary conditions are assumed. The euclidean time extent, however, is taken infinite. This choice implies an exponential decay of two-point functions and  simplifies our calculations. Still, this simplification is a good approximation for many lattice QCD simulations. Another simplification concerns the masses of the up and down type quarks which we assume to be equal. Consequently, all three pions as well as the nucleons (proton and neutron) are mass degenerate.\n\nWe are interested in the two-point correlation functions of a nucleon interpolating field $N$ with definite parity, \n\\begin{eqnarray}\nC_{\\pm}(t)& =& \\int_{L^3} {\\rm d}^3{{x}}\\, \\langle  N_{\\pm}(\\vec{x},t) \\overline{N}_{\\!\\pm}(0,0)\\rangle\\ .\\label{DefNCorr}\n\\end{eqnarray}\nHere we defined $N_{\\pm} = \\Gamma_{\\pm} N$ and $\\overline{N}_{\\!\\pm} = \\overline{N} \\Gamma_{\\pm}$ with the standard projectors \n$\\Gamma_{\\pm} = (\\gamma_0\\pm 1)\/2$.\\footnote{Note that in order to keep the notation simple we often suppress the Dirac and\/or flavor indices. For example, eq.\\ \\pref{DefNCorr} contains an implicit summation over the Dirac indices.} $N$ itself is an interpolating field with the quantum numbers of the nucleon. Various choices are possible and we discuss concrete examples in the next subsection. For the moment we do not need to specify $N$ any further. \n\nLet us consider the positive parity correlator.\nThe integration over the spatial volume in Eq.~\\pref{DefNCorr} projects on states with zero total momentum. Hence, in the spectral decomposition of the correlator for large euclidean times $t\\gg0$ the dominant contribution comes from the single-particle state with the particle being the nucleon at rest, \n\\begin{equation}\n\\label{spcontr}\nC_{+, N}(t)=\n\\frac{1}{2M_{\\pm}}\\;|\\langle 0|N_{+}(0)|N({\\vec p}=0)\\rangle|^2 e^{-M_{N} t } \\,.\n\\end{equation}\nHere $|N({\\vec p}=0)\\rangle$ is the nucleon state \nand $M_N$ denotes the nucleon mass.\n\nThe interpolating field excites other states with the same quantum numbers as well. The contribution of an excited nucleon has the same form as Eq.~(\\ref{spcontr}) with the appropriate mass $M'> M_{N}$. In addition we expect contributions from multi-hadron states. For sufficiently small pion masses the dominant multi-hadron states are those containing additional pions. For the two-particle nucleon-pion state contribution one finds\n\\begin{eqnarray}\n\\label{tpcontr}\nC_{+, N\\pi}(t)&=&\\frac{1}{L^3}\\;\\sum_{{\\vec p}}\\frac{1}{4E_{N} E_{\\pi}}\\,\n|\\langle 0|N_{+}(0)|N({\\vec p}) \\pi(-{\\vec p})\\rangle|^2 e^{-E_{\\rm tot}t}\\,.\n\\end{eqnarray}\n$E_{\\rm tot}$ is the total energy of the state and $E_N$, $E_{\\pi}$ are the individual energies of the nucleon and the pion, respectively. For weakly interacting pions $E_{\\rm tot}$ equals approximately the sum $E_N+E_{\\pi}$. The sum over momenta runs over all momenta allowed by the boundary conditions imposed for the finite spatial volume, e.g.\\ ${\\vec p}=2\\p{\\vec n}\/L$ with ${\\vec n}$ having integer-valued components. \n\nA simple dimensional analysis can be employed to make the volume suppression more quantitative. \nAssume the interpolating fields are local 3-quark-operators without derivatives.\nIn that case the mass dimension of the matrix elements in \\pref{spcontr} and \\pref{tpcontr} are 7\/2 and 5\/2, respectively. Making the naive assumption $\\langle 0|N_{\\pm}(0)|N({\\vec p}) \\pi(-{\\vec p})\\rangle\\approx \\langle 0|N_{\\pm}(0)|N_{\\pm}({\\vec p}=0)\\rangle \/ f_{\\pi}$ we can estimate the ratio of the two-particle and one-particle contributions as\n\\begin{equation}\\label{NaiveEst}\n\\frac{C_{+, N\\pi}(t)}{C_{+,N}(t)}\\approx \\frac{1}{2(f_{\\pi}L)^2M_{\\pi}L}\\frac{M_{\\pi}}{E_{\\pi}}\\frac{M_{N}}{E_{N}}e^{-(E_{\\rm tot} - M_{N})t}\\,.\n\\end{equation}\nIf we assume the values $M_{\\pi}\\approx 200$ MeV and $L\\approx 4$ fm we roughly find $[2(f_{\\pi} L)^2M_{\\pi}L]^{-1}\\approx 1\/30$. The additional factors suppress the two-particle state contribution further, so we expect its contribution to the correlator to be rather small. \n\nStates with more than one pion contribute analogously to \\pref{tpcontr}, but each additional pion contributes an additional factor $[2(f_{\\pi} L)^2M_{\\pi}L]^{-1}$, i.e.\\ the more pions in the state the larger the suppression of its contribution with the spatial volume.\n\nOur discussion applies to the negative parity correlator as well. In this channel the lightest single-particle state is, in infinite volume,  the $N^*(1535)$. However, the state with the lowest energy is the nucleon-pion state with both particles at rest, provided the pion mass is sufficiently small. Thus the two-particle state dominates the long time behaviour and the ratio analogous to the one in \\pref{NaiveEst} will diverge for $t\\rightarrow \\infty$ in the negative parity channel. \n\n\\subsection{Interpolating fields for the nucleon}\\label{ssect:QCDinterpolaters}\n\nAs already mentioned, there exist many choices for the interpolating field (``operator'') $N$ with the quantum numbers of the nucleon. \nThe number is significantly reduced if we consider local operators composed of three quark fields at the same point $x$. If, in addition, we constrain ourselves to operators  without derivatives there exist only five different ones. As a consequence of Fierz identities only two are independent \\cite{Ioffe:1981kw,Espriu:1983hu} (see also \\cite{Nagata:2008zzc}) and we focus on those. In order to write them down it is convenient to introduce the quark field doublet ${\\tilde{q}}$ as \n\\begin{equation}\\label{qtilde}\n{\\tilde{q}}=q^{\\rm T} C \\gamma_5 (i\\sigma_2)\\,.\n\\end{equation}\nHere $q= (u,d)^{\\rm T}$ is the isospin doublet of the quark fields, $C$ denotes the Dirac spinor charge conjugation matrix satisfying $\\gamma_{\\mu}^{\\rm T} = -C\\gamma_{\\mu} C^{-1}$, and $\\sigma_2$ is the second (isospin) Pauli matrix. With these definitions the two nucleon operators can be written as\n\\begin{equation}\\label{DefN12}\n\\begin{array}{l}\n N_1 \\,= \\, ({\\tilde{q}} q) q\\,,\\\\[0.4ex]\n N_2\\,=\\, ({\\tilde{q}} \\gamma_5 q) \\gamma_5q\\,.\n \\end{array}\n\\end{equation}\nThis compact form suppresses the contraction of the isospin and Dirac indices in the bilinear quark fields $({\\tilde{q}} q)$ and $({\\tilde{q}} \\gamma_5 q)$ (``diquarks'') and the summation over the color indices with an $\\epsilon_{abc}$ to form a color singlet. The nucleon operators $N_i$ are still isospin doublets. To project onto the quark content of the proton and neutron we need to contract with the isospin basis vectors $e_p=(1,0)^{\\rm T}$ and $e_n=(0,1)^{\\rm T}$, respectively. However, in our case with preserved isospin symmetry any unit vector would be equally good.\n\nIn the next section we need the counterparts of the nucleon operators $N_i$ in ChPT. The mapping follows the standard procedure and rests on the transformation properties of $N_i$ under chiral and parity transformations. \nThe transformation properties under (singlet and non-singlet) chiral transformations have been studied in detail in Ref.\\ \\cite{Nagata:2008zzc}. Here we simply summarize the relevant results. \n\nWe decompose the quark fields into right- and left-handed components, $q=q_R+q_L$, with the usual chiral projectors $P_{+}=(1+\\gamma_5)\/2$ and $P_{-}=(1-\\gamma_5)\/2$. It then follows that the field in \\pref{qtilde} also decomposes according to ${\\tilde{q}}={\\tilde{q}}_R+{\\tilde{q}}_L$, with ${\\tilde{q}}_R={\\tilde{q}} P_+$ and ${\\tilde{q}}_L={\\tilde{q}} P_{-}$. The group of non-singlet chiral transformations is $G=SU(2)_R\\otimes SU(2)_L$, and under transformations $R\\otimes L\\in G$ the chiral quark fields transform according to\n\\begin{equation}\\label{chiralTrafoq}\n\\begin{array}{rclcl}\nq & =&  q_R + q_L &  \\xrightarrow{\\,R,L\\,} & R \\,q_R + L  \\,q_L\\,,\\\\[0.4ex]\n\\tilde{q} & =&  \\tilde{q}_R + \\tilde{q}_L & \\xrightarrow{\\,R,L\\,} & \\tilde{q}_R \\,R^{\\dagger}  + \\tilde{q}_L \\,L^{\\dagger}\\,.\n\\end{array}\n\\end{equation}\nThe diquarks decompose into ${\\tilde{q}} q ={\\tilde{q}}_R q_R + {\\tilde{q}}_Lq_L$  and ${\\tilde{q}} \\gamma_5 q={\\tilde{q}}_R q_R - {\\tilde{q}}_Lq_L$, hence they transform as singlets under chiral transformations. Consequently, the transformation behavior of the nucleon operators is determined by the third quark field contribution $q$ and $\\gamma_5 q$, given in \\pref{chiralTrafoq}. Decomposing this quark field into right- and left-handed components the complete nucleon fields $N_{1,2}$ itself can be written as a sum of a right-handed and a left-handed term with the following transformation behavior under chiral transformations:\n\\begin{equation}\\label{NTrafoLR}\n\\begin{array}{rclcl}\nN_{i} & =& N_{i,R} +N_{i,L} &  \\xrightarrow{\\,R,L\\,} & RN_{i,R} +LN_{i,L}\n\\end{array}\n\\end{equation}\n Concerning parity one finds that ${\\tilde{q}} q$ and ${\\tilde{q}} \\gamma_5 q$ transform as a scalar and a pseudo scalar, respectively. Thus, both $N_1$ and $N_2$ transform as a Dirac spinor under parity, $N_i \\rightarrow \\gamma_0 N_i$.\n\\begin{equation}\\label{NTrafoP}\n\\begin{array}{rclcl}\nN_{i} & =& N_{i,R} +N_{i,L} &  \\xrightarrow{\\, P \\,} & \\gamma_0(N_{i,L} +N_{i,R})\\,.\n\\end{array}\\end{equation}\n\nSo far we considered local interpolating fields only. In lattice QCD so-called smeared interpolators are very often used, mainly to suppress excited-state contributions in the correlation function. Smeared nucleon interpolating fields are formed as in \\pref{DefN12} but with the local quark fields replaced by smeared ones, which are generically of the form\\footnote{We use a continuum notation here. In lattice QCD the integral is replaced by a sum over the lattice points.} \n\\begin{equation}\nq_{\\rm sm} (x) = \\int {\\rm d^4}y K(x - y) q(y)\n\\end{equation}\nwith some gauge covariant kernel $K(x-y)$ which is essentially zero for $|x-y|$ larger than some ``smearing radius'' $R$. The kernel depends on the details of the smearing procedure. Gaussian and exponential smearing \\cite{Gusken:1989ad,Gusken:1989qx,Alexandrou:1990dq} is local in time and the kernel contains a delta function in the euclidean time coordinate. In contrast, the gradient flow \\cite{Luscher:2013cpa} is a truly four-dimensional smearing. \n\nWhat matters here are the transformation properties of the smeared quark fields. Provided the kernel is diagonal in spinor space (as it is for Gaussian smearing and the gradient flow) the smeared quark fields transform just as the unsmeared ones under parity and global chiral transformations.  Consequently, also the nucleon interpolating fields formed with the smeared quark fields transform according to \\pref{NTrafoLR} and \\pref{NTrafoP}, just as their local counterparts. Since the symmetry properties of the interpolating fields essentially determine their expression in ChPT we can already conclude that both local and smeared interpolating fields are mapped onto the same effective operator, differing in their values for the LECs only. We come back to this issue in section \\ref{sseceffFields}. \n\nThe two interpolating fields in eq.\\  \\pref{DefN12} were originally discussed in the context of QCD sum rule calculations, and in lattice QCD simulations the discretized version of these continuum interpolaters are used.  An alternative approach for the construction of baryonic operators starts directly from the irreducible representations of the cubic group of the space-time lattice \\cite{Basak:2005aq}. In order to discuss this type of operators one first has to perform a mapping to the Symanzik effective theory, the leading part being continuum QCD followed by corrections proportional to powers of the lattice spacing \\cite{Symanzik:1983dc}. To the lattice operators whose leading Symanzik term is given by the interpolating fields in \\pref{DefN12} the following discussion equally applies up to corrections proportional to the lattice spacing.\\footnote{Lattice artifacts are of course also present in lattice simulations that use the discretized expressions of $N_1$ or $N_2$.}\n\nFinally, non-relativistic interpolators can be constructed from the relativistic ones if the standard non-relativistic (Dirac-)representation for the $\\gamma$-matrices is used \\cite{Billoire:1984jm}. In appendix \\ref{appendixA} we discuss briefly the correlation function of the non-relativistic limit of $N_1$, which is rather easily obtained from the result using the relativistic $N_1$.\n\n\n\n\\section{\\label{secChPT} The nucleon two-point correlators in ChPT}\n\\subsection{\\label{ssecLag} The chiral Lagrangian}\nThe framework for our calculations is covariant baryon chiral perturbation theory \\cite{Gasser:1987rb,Becher:1999he}.\\footnote{A thorough and pedagogical introduction to the subject can be found in Ref.\\ \\cite{Scherer:2012xha}, for example.} In this section we summarize a few relevant formulae since we work in euclidean space time and most references assume the Minkowski metric.\n\nWe consider the chiral effective Lagrangian\\footnote{The superscripts denote the low-energy dimensions of these lagrangians, i.e.\\ they count the number of derivatives  and the power of quark mass terms \\cite{Gasser:1987rb}.}\n\\begin{equation}\\label{effLag}\n{\\cal L}_{\\rm eff}={\\cal L}_{N\\pi}^{(1)} + {\\cal L}_{\\pi\\pi}^{(2)}\\,.\n\\end{equation}\nHere ${\\cal L}_{\\pi\\pi}^{(2)}$ is the standard two-flavor mesonic chiral Lagrangian to leading order \\cite{Gasser:1983ky,Gasser:1983yg}. According to the conventions used here it reads\n\\begin{equation}\n{\\cal L}_{\\pi\\pi}^{(2)} = \\frac{f^2}{4} {\\rm Tr}[\\partial_{\\mu}U\\partial_{\\mu}U^{\\dagger}] +\\frac{f^2B}{2}{\\rm Tr}[{\\cal M}(U+U^{\\dagger})]\\,.\n\\end{equation}\n$f,B$ are the standard LO LECs related to the pion decay constant and chiral condensate in the chiral limit.\\footnote{Our conventions correspond to $f_{\\pi}= 92.2$~MeV.} The pion fields are contained in the field $U$ according to\n\\begin{equation}\nU(x) = \\exp\\left[\\frac{i}{f}\\pi^a(x)\\sigma^a\\right],\n\\end{equation}\nwith the usual Pauli matrices $\\sigma^a$. ${\\cal M}$ denotes the quark mass matrix. With equal quark masses $m$ for the up and down quark it is proportional to the unit matrix. In that case all three pions have the same mass which to LO is related to the quark mass via $M_{\\pi}^2= 2Bm$. \n\nThe second part in the chiral lagrangian \\pref{effLag} contains the nucleon fields and their coupling to the pions,\n\\begin{equation}\\label{LNpi}\n{\\cal L}_{N\\pi}^{(1)}=\\overline{\\Psi} \\Big(\\slashed{D}+M_N -i \\frac{g_A}{2}\\slashed{u}\\gamma_5\\Big)\\Psi\\,.\n\\end{equation}\nThe  fields $\\Psi=(p,n)^T$ and $\\overline{\\Psi}=(\\overline{p},\\overline{n})$ \ndenote the nucleon fields with two Dirac spinors for the proton $p$ and the neutron $n$. \n$M_N$ and $g_A$ are the nucleon mass and the axial-vector coupling constant in the chiral limit. Since we assume isospin symmetry the proton and the neutron are mass degenerate. \n\nThe pion fields enter ${\\cal L}_{N\\pi}^{(1)}$ via the field $u_{\\mu}$, the so-called {\\em chiral vielbein}, defined by\n\\begin{equation}\nu_{\\mu} = i[u^{\\dagger}\\partial_{\\mu}u - u \\partial_{\\mu}u^{\\dagger}],\\quad u(x)\\,=\\,\\sqrt{U(x)}\\,.\n\\end{equation}\nA second source of pion-nucleon coupling stems from the covariant derivative $\\slashed{D}=\\gamma_{\\mu}D_{\\mu}$ in \\pref{LNpi}, with\n$ D_{\\mu}\\Psi = \\Big(\\partial_{\\mu} + \\Gamma_{\\mu}\\Big)\\Psi$ and \n $\\Gamma_{\\mu}=\\left[u^{\\dagger}\\partial_{\\mu}u + u \\partial_{\\mu}u^{\\dagger}\\right]\/2$. \n\n\\begin{table}[tbdp]\n\\begin{center}\n\\begin{tabular}{c|cccccc}\n\\hline\\hline\n&\\,&  $\\Psi$&\\,\\,\\, & $u$ & \\,\\,\\,& $u_{\\mu}$\\\\ \n\\hline\n\\,$R\\otimes L$ \\, &&$K\\Psi$  && $RuK^{\\dagger}  = K u L^{\\dagger} $ && $K u_{\\mu} K^{\\dagger} $\\\\\n$P$  && $\\gamma_0\\Psi$ && $u^{\\dagger}$ &&$ (- 1)^{\\delta_{\\mu 0}} u_{\\mu} $\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{tab:SymFields}Transformation behavior of the nucleon and pseudo scalar fields under chiral and parity transformations (see Ref.\\ \\cite{Wein:2011ix}). The SU(2) matrix $K$ appearing in the first row is defined by the transformation law of $u$ such that $u^2=U$ transforms in the standard way.}\n\\end{table}\n\nThe construction of the chiral lagrangian is based on the symmetry properties of the underlying QCD lagrangian, which ${\\cal L}_{\\rm eff}$ needs to reproduce. The transformation behavior of the nucleon and pseudo scalar fields under chiral and parity transformations is briefly summarized in table \\ref{tab:SymFields} (for details see Ref.\\ \\cite{Scherer:2012xha}, for example).\n\nExpanding $u_{\\mu}$ and $\\Gamma_{\\mu}$  we obtain pion-nucleon interaction terms with various numbers of pion fields.  Since $u_{\\mu}$ is parity-odd and $\\Gamma_{\\mu}$ is parity-even the leading interaction term with one pion field only stems from $u_{\\mu}$ and reads\n\\begin{equation}\\label{Lintexpand}\n{\\cal L}_{\\rm int, LO}^{(1)} = \\frac{ig_A}{2f}\\overline{\\Psi}\\gamma_{\\mu}\\gamma_5\\sigma^a \\Psi \\, \\partial_{\\mu} \\pi^a\\,.\n\\end{equation}\nThis interaction term couples two axial vectors to obtain a Lorentz scalar. In addition, isospin symmetry is preserved. \n\nThe chiral Lagrangian incorporates a derivative expansion and the chiral dimension counts the number of derivatives  and powers of the quark mass. The complete list of terms through fourth order is given in \\cite{Fettes:2000gb}. For the purpose of this paper, however,  the term in \\pref{Lintexpand} is sufficient.\n\nFor the perturbative calculation in section \\ref{ssecNpiCont} we need the propagators for the nucleon and the pion in position space.  The  pion propagator is the same as in Ref.\\ \\cite{Bar:2012ce},\n\\begin{equation}\\label{scalprop}\nG^{ab}(x,y)=   \\delta^{ab}L^{-3}\\sum_{\\vec{p}} \\frac{1}{2 E_{\\pi}} e^{i\\vec{p}(\\vec{x}-\\vec{y})} e^{-E_{\\pi} |x_0 - y_0|}\\,,\n\\end{equation} \nwith \npion energy $E_{\\pi} =\\sqrt{\\vec{p}^2 +M_{\\pi}^2}$. The nucleon propagator $S^{ab}_{\\alpha\\beta}(x,y)$ is also easily derived from the quadratic term in \\pref{LNpi}, and it reads\n\\begin{equation}\\label{NucleonProp}\nS_{\\alpha\\beta}^{ab}(x,y)=   \\delta^{ab} L^{-3}\\sum_{\\vec{p}} \\frac{Z_{p,\\alpha\\beta}^{\\pm}}{2E_N} e^{i\\vec{p}(\\vec{x}-\\vec{y})} e^{-E_N |x_0 - y_0|}\\,.\n\\end{equation}\n$a,b$ and $\\alpha\\beta$ refer to the isospin and Dirac indices, respectively. The factor $Z^{\\pm}_{\\vec{p}}$  (spinor indices suppressed) in the numerator is defined as\n\\begin{equation}\nZ_{\\vec{p}}^{\\pm}=-i\\vec{p}\\cdot\\vec{\\gamma} \\pm E_N \\gamma_0+M_N\\,, \n\\end{equation}\nwhere the $+$ ($-$) sign applies to $x_0 > y_0$ ($x_0 < y_0$), and the nucleon energy $E_N=\\sqrt{\\vec{p}^2 +M_N^2}$ involves the nucleon mass. \nThe sum in both propagators runs over the discrete spatial momenta that are compatible with periodic boundary conditions, i.e.\\ \n${\\vec p}=2\\p{\\vec n}\/L$  with ${\\vec n}$ having integer-valued components.\n\n\\subsection{\\label{sseceffFields} The chiral expansion of the interpolating fields}\n\nThe construction of the nucleon operators in baryon ChPT follows the standard procedure.  \nBased on the symmetry properties of the operators on the quark level we write down the most general expression in the effective theory that has the same symmetry properties. This has essentially been done in Ref.\\ \\cite{Wein:2011ix} and we summarize the results needed in the following.\\footnote{Instead of $N_{1,2}$ two other operators, related to $N_{1,2}$ by Fierz identities, are considered in Ref.\\ \\cite{Wein:2011ix}. However, the chiral expansion is essentially the same.} \n\nThe nucleon operators in the effective theory needs to transform as given in \\pref{NTrafoLR} and \\pref{NTrafoP} under chiral and parity transformations. Basically, the nucleon operators are a sum of a right- and left-handed spinor and can be written as (we follow closely the notation introduced in Ref.\\ \\cite{Wein:2011ix})\n\\begin{equation}\\label{Neff}\nN= \\sum_{n}\\sum_k^{i_{n}} \\alpha_k^{(n)} \\left(N^{(n)}_{k,R} +  N^{(n)}_{k,L}\\right)\\,.\n\\end{equation}\n$N^{(n)}_{k,R}$ and  $N^{(n)}_{k,L}$ are operators with low-energy dimension $n$. $i_n$ denotes the number of operators with chiral dimension $n$, which are labelled by the index $k$.  Under chiral and parity transformations the fields in \\pref{Neff} transform according to \n\\begin{equation}\\label{NeffTrafo}\n\\begin{array}{rcl}\nN^{(n)}_{k,R} +N^{(n)}_{k,L}    &  \\xrightarrow{\\,R,L \\,}  & R\\, N^{(n)}_{k,R} +L\\, N^{(n)}_{k,L} ,\\\\[0.4ex]\nN^{(n)}_{k,R} +N^{(n)}_{k,L}    &  \\xrightarrow{\\,\\,\\,P \\,\\,\\,}  & \\gamma_0 (N^{(n)}_{k,L} + N^{(n)}_{k,R})\\, . \n\\end{array}\n\\end{equation}\nEach term on the right hand side of \\pref{Neff} comes with its own LEC $\\alpha_k^{(n)}$, and it is parity that relates the coefficients of the right- and left-handed contributions. \n\nAn incomplete list of operators through chiral dimension two can be found in Ref.\\ \\cite{Wein:2011ix}. For convenience we reproduce the ones through $n=1$ in table \\ref{table:Nop}. It is straightforward to check that these operators satisfy the transformation laws given in \\pref{NeffTrafo}.\n\n\\begin{table}[tbdp]\n\\begin{center}\n\\begin{tabular}{ccccccc}\n\\hline\\hline\n$n$ &\\,& $k$&\\,\\,\\, & $N^{(n)}_{k,R}$ & \\,\\,\\,& $N^{(n)}_{k,L}$\\\\ \n\\hline\n0 &&1 && $u P_+ \\Psi$ && $u^{\\dagger} P_- \\Psi$\\\\\n1 && 1 && $u u_{\\mu} P_+ \\gamma_{\\mu} \\Psi$ &&$ - u^{\\dagger} u_{\\mu} P_- \\gamma_{\\mu} \\Psi $\\\\\n1 && 2 && $u u_{\\mu} P_+  D_{\\mu} \\Psi $&&$ - u^{\\dagger} u_{\\mu} P_- D_{\\mu} \\Psi$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{table:Nop}Low-energy operators for the nucleon interpolating fields through chiral dimension one (see Ref.\\ \\cite{Wein:2011ix}).}\n\\end{table}\n\nNote that $N$ in \\pref{Neff} does not carry an index $i$ that would refer explicitly to one of the two interpolating fields defined in \\pref{DefN12}. We dropped this index because the chiral expansion for both operators is the same due to their similar transformation behavior. The only difference are different values for the LECs in the chiral expansion. In order to keep our notation simple we suppress an additional index at the operator and the LECs in the following. \n\nSimilarly, for the smeared interpolating fields discussed at the end of section \\ref{ssect:QCDinterpolaters} we also find the same effective operator \\pref{Neff} with different LECs. However, one qualification has to be made. Smeared interpolators with some ``size'' are mapped onto the pointlike nucleon field in the chiral effective theory. For this to be a good approximation the smearing radius needs to small compared to the Compton wave length of the pion. Provided this condition is met the pions do not distinguish between smeared and pointlike interpolating fields.\\footnote{A concrete example is given in Ref.\\ \\cite{Bar:2013ora} where ChPT for some observables based on the gradient flow has been constructed.}\n\n\n\nAt lowest low-energy dimension only one operator contributes. Expanding $u,u^{\\dagger}$ in powers of pion fields and keeping only the terms up to linear order we obtain for $N$ the expression\n\\begin{equation}\\label{Neffexp}\nN(x)=  \\tilde{\\alpha} \\left(\\Psi(x) + \\frac{i}{2f} \\pi(x) \\gamma_5\\Psi(x)\\right)\\,,\\qquad \\tilde{\\alpha}\\,=\\,4\\alpha_0^{(0)}\\,.\n\\end{equation}\nThe first LO term is proportional to the nucleon field $\\Psi$, as expected. The second NLO term (suppressed by $1\/f$) involves a nucleon-pion coupling that will contribute to the two-particle nucleon-pion terms \\pref{tpcontr} in the nucleon correlation function. \n\n\\subsection{\\label{ssecNpiCont} Perturbative expansion of the correlation functions}\n\n\\begin{figure}[tp]\n\\begin{center}\n\\includegraphics[scale=0.4]{Figures\/fig1a}\\\\\na)\\\\[3ex]\n\\includegraphics[scale=0.4]{Figures\/fig1b}\\\\\nb)\\\\[3ex]\n\\includegraphics[scale=0.4]{Figures\/fig1c}\\hspace{1cm}\\includegraphics[scale=0.4]{Figures\/fig1d}\\\\\nc)\\hspace{5cm} d)\\\\[3ex]\n\\includegraphics[scale=0.4]{Figures\/fig1e}\\\\\ne)\n\\caption{Feynman diagrams for the nucleon correlation function. The squares represent the nucleon operator at times $t$ and $0$, where the open and solid squares denote the leading and next-to-leading order terms given in \\pref{Neffexp}. The circles represent a vertex insertion at an intermediate space time point; and an integration over this point is implicitly assumed. The solid and dashed lines represent nucleon and pion propagators, respectively. }\n\\label{fig:diagrams}\n\\end{center}\n\\end{figure}\n\nWe are now in the position to compute the correlation functions \\pref{DefNCorr} perturbatively within the chiral effective theory. The leading contribution is obtained by taking into account the LO term in \\pref{Neffexp} for $N$ and $\\overline{N}=N^{\\dagger} \\gamma_0$.   Since these fields are proportional to the nucleon fields $\\Psi,\\overline{\\Psi}$ the LO contribution is essentially the nucleon propagator. In terms of Feynman rules in position space this contribution is represented by the Feynman diagram in figure \\ref{fig:diagrams}a. Taking into account \\pref{NucleonProp} for the nucleon propagator the LO results for the correlators are easily obtained, \n\\begin{eqnarray}\\label{CorrLO}\nC_{+,N}(t)&=& 2|\\tilde{\\alpha}|^2 e^{-M_N t}\\,,\\qquad C_{-,N}(t)\\,=\\, 0\\,.\n\\end{eqnarray}\nThese results are a single-particle state contributions to the correlation function, and by comparing with \\pref{spcontr} we can read off the LO relation between the vacuum-to-nucleon matrix element and the LEC $\\tilde{\\alpha}$,\n\\begin{equation}\n|\\langle 0|N_{+}(0)|N_{+}({\\vec p}=0)\\rangle|^2 = 4M_N |\\tilde{\\alpha}|^2\\,.\n\\end{equation}\n$C_{-,N}$ in \\pref{CorrLO} vanishes at this order because our effective theory does not contain the negative parity nucleon as a degree of freedom. \n\nThe diagrams in Figs.\\ \\ref{fig:diagrams}b - \\ref{fig:diagrams}e form the leading contribution to the two-particle nucleon-pion part of the correlation function. The dashed line represents the pion propagator, which, together with the nucleon propagator, leads to terms with the expected exponential fall-off with $E_{\\rm tot} = E_N+E_{\\pi}$.\\footnote{Fig.\\ \\ref{fig:diagrams}e also contains a contribution with  a time dependence proportional to $t\\exp(-M_N t)$. This results in the renormalization of the nucleon mass and can be ignored for our purposes.} \n\nThe calculation of the diagrams \\ref{fig:diagrams}b - \\ref{fig:diagrams}e is straightforward, and the final results can be compactly written as \n\\begin{eqnarray}\nC_{+,{N\\pi}}(t)& = &2|\\tilde{\\alpha}|^2\\frac{3}{8(fL)^2 mL} \\sum_{\\vec{p}} \n\\frac{m}{E_{\\pi}} \\frac{E_N-M_N}{2 E_N}\\left[1 -g_A \\frac{E_{\\rm tot}+M_N}{E_{\\rm tot}-M_N}\\right]^2\n e^{-E_{\\rm tot} |t|}\\,,\\label{Cnpplus}\\\\\nC_{-,{N\\pi}}(t)& = &2|\\tilde{\\alpha}|^2\\frac{3}{8(fL)^2 mL} \\sum_{\\vec{p}} \n\\frac{m}{E_{\\pi}}\\frac{E_N+M_N}{2E_N}\\left[1 -g_A \\frac{E_{\\rm tot}-M_N}{E_{\\rm tot}+M_N}\\right]^2\n e^{-E_{\\rm tot}|t| }\\,.\\label{Cnpminus}\n\\end{eqnarray}\nNote that the $\\vec{p}=0$ term  in $C_{+,{N\\pi}}(t)$ correctly vanishes, as required by parity.\n\nThe overall factor $2|\\tilde{\\alpha}|^2$ has its origin in the appearance of $|\\tilde{\\alpha}|^2$ as an overall factor for the two terms in \\pref{Neffexp}. This implies that the relative size of the two-particle state contributions in \\pref{Cnpplus}, \\pref{Cnpminus} and the one-particle state contribution in \\pref{CorrLO} does not contain the LEC $\\tilde{\\alpha}$ associated with the nucleon interpolating field. It only depends on the LECs $f$ and $g_A$ of the effective action. However, this is true to LO only. Taking into account the higher order operators in table \\ref{table:Nop} their LECs will not cancel in the ratio. \n\nRecall that the chiral expansion for the nucleon operators $N_1$ and $N_2$ are the same, the only difference being different LECs. Restoring the label $i=1,2$ in the LEC $\\tilde{\\alpha}$ the results given above (with $|\\tilde{\\alpha}|^2$ replaced by $|\\tilde{\\alpha}_i|^2$) refer to the correlation functions where the same operator is used for both source and sink. In case of the correlator with $N_1(x)$ and $\\overline{N}_2(0)$ we simply need to replace $|\\tilde{\\alpha}|^2$ by $\\tilde{\\alpha}_1{\\tilde{\\alpha}_2^{*}}$. Since the same combination appears as an overall factor in the single-particle and two-particle state contributions it still drops out in the ratio of these two contributions. The same statement applies to smeared operators.\n\nThe summation in \\pref{Cnpplus}, \\pref{Cnpminus} is  over all lattice momenta compatible with periodic boundary conditions. Those that are related by the symmetries of the spatial lattice lead to the same contribution, hence it is convenient to  sum over the absolute value $p=|\\vec{p}|$. Imposing periodic boundary conditions the absolute value can assume the values $p_n=(2\\pi\/L)\\sqrt{n}$, $n\\equiv n_1^2+n_2^2+n_3^2$, with the $n_k$ being integers, and the sums in the results given above are replaced according to\n\\begin{equation}\\label{redsum}\n\\sum_{\\vec{p}} \\longrightarrow \\sum_{p_n} m_n\\,.\n\\end{equation}\nThe multiplicities $m_n$ count the number of vectors $\\vec{p}$ with the same $p_n$. Multiplicities for $n\\leq 20$ are given in Ref.\\ \\cite{Colangelo:2003hf} (for convenience we summarize the first eight in table \\ref{tabledn}).\n\n\\begin{table}[tbdp]\n\\begin{center}\n\\begin{tabular}{lrrrrrrrrr}\n\\hline\\hline\n$n$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\\\ \n$m_n$ & \\phantom{1}1 & \\phantom{1}6 & 12 & \\phantom{1}8 & \\phantom{1}6 & 24 & 24 & \\phantom{0}0 & 12\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{\\label{tabledn}Multiplicities $m_n$ in eq.\\ \\pref{redsum} for $n\\le 8$ (see Ref.\\ \\cite{Colangelo:2003hf}).}\n\\end{table}\n\n\\subsection{\\label{ssecResults} Final Results}\n\nAdding the two results in \\pref{CorrLO} and \\pref{Cnpplus} the positive parity correlation function can be written as\n\\begin{equation}\\label{Cplusfinal}\nC_{+}(t)= 2|\\tilde{\\alpha}|^2 e^{-M_N t}\\left[ 1+ \\sum_{p_n} c^{+}_n e^{-(E_{{\\rm tot},n} -M_N) t}\\right]\\,,\n\\end{equation}\nwhere we introduced new dimensionless coefficients \n\\begin{eqnarray}\nc^{+}_n &=& \\frac{3m_n}{8(fL)^2 E_{{\\pi,n}}L} h^{+}_n\\,,\\label{cnp}\\\\\n  h^{+}_n &=&  \\frac{E_{N,n}-M_N}{2 E_{N,n}}\\left[1 -g_A \\frac{E_{{\\rm tot},n}+M_N}{E_{{\\rm tot},n}-M_N}\\right]^2\\,.\\label{hnp}\n\\end{eqnarray}\nResult \\pref{Cplusfinal} corresponds to the example we discussed briefly in the introduction. The coefficients $c_n^+$ are equal to $b_n\/b_0$ and all nucleon-pion state contributions are given. \n\nWe already mentioned that the positive parity two-point correlator was independently calculated in Ref.\\ \\cite{Tiburzi:2015tta} using HBChPT \\cite{Georgi:1990um,Jenkins:1990jv}. In that non-relativistic formulation one drops the antinucleon degrees of freedom and the dispersion relation of the heavy nucleon is non-relativistic. If we expand $E_{N,n}\\approx M_N+ p_n^2\/2M_N$ in \\pref{hnp} and drop all but the dominant terms we find $h_n^+ \\approx g_A^2 p_n^2\/E_{\\pi}^2$ and reproduce the result in \\cite{Tiburzi:2015tta}.  \n\nThe negative parity channel is slightly different since there is no  single particle state contribution stemming from the nucleon. The leading single-particle state contribution comes from the negative parity partner $N^*$ which is not a degree of freedom in our effective theory. And even if we included it explicitly the coupling of the interpolating field $N^-$ to the $N^*$ would come with a LEC unrelated to the LEC $\\tilde{\\alpha}$ that enters result \\pref{Cnpminus}. \n\nThe dominant contribution in our result for $C_-$ stems from the nucleon-pion state with the nucleon and the pion at rest. Taking this contribution out of the sum we arrive at the form\n\\begin{equation}\\label{Cminusfinal}\nC_{-}(t)= 2|\\tilde{\\alpha}|^2 \\frac{3}{8(fL)^2 M_{\\pi} L} e^{-(M_N + M_{\\pi}) t}\\left[ 1+ \\sum_{p_{n}\\neq 0} c^{-}_n \\,e^{-(E_{{\\rm tot},n} -M_N - M_{\\pi}) t}\\right]\\,,\n\\end{equation}\nwith a coefficient \n\\begin{eqnarray}\nc^{-}_n &=& m_n\\frac{M_{\\pi}}{E_{\\pi,n}} h^{-}_n\\,,\\label{cnm}\\\\\nh^{-}_n &=&  \\frac{E_{N,n}+M_N}{2 E_{N,n}}\\left[1 -g_A \\frac{E_{{\\rm tot},n}-M_N}{E_{{\\rm tot},n}+M_N}\\right]^2\\,.\n\\end{eqnarray}\n\nNote that the results for the two parity channels are proportional to the unknown LEC $|\\tilde{\\alpha}|^2$. Thus, taking the ratio $C_{-}(t)\/C_{+}(t)$ this constant drops out and the only LECs contributing to this ratio are $f$ and $g_A$ of the LO chiral lagrangian. \n\n\n\n\\subsection{\\label{ssecNumEst} Numerical estimates}\n\nBefore trying to estimate the impact of the nucleon-pion-state contribution to the two-point function it is necessary to discuss the conditions for the applicability of the results derived in the last section. \n\nChPT is an expansion in the pion mass and momentum. Both need to be small compared to the chiral symmetry breaking scale $\\Lambda_{\\chi}$, which is typically identified with $4\\pi f_{\\pi}$. In a finite spatial volume with periodic boundary conditions the pion momenta are discrete and the condition for the applicability of the chiral expansion reads \\cite{Colangelo:2003hf}\n\\begin{equation}\\label{boundpn}\n\\frac{p_n}{\\Lambda_{\\chi}} = \\frac{M_{\\pi}}{2f_{\\pi}} \\frac{\\sqrt{n}}{M_{\\pi} L}  \\ll 1\\,.\n\\end{equation}\nEven though the pion mass cancels on the right hand side we prefer this form since lattice QCD configurations are often characterized in terms of the pion mass and $M_{\\pi}L$. Given these two numbers and the pion decay constant $f_{\\pi}\\approx 90$MeV eq.\\ \\pref{boundpn} provides a bound on the pion momenta and the label $n$. \nTable \\ref{tab:expansion} lists a few representative values that approximately match the parameters in present-day lattice simulations.\\footnote{For the calculation of the light hadron spectrum in \\cite{Durr:2008zz} the BMW collaboration generated a lattice ensemble with $M_{\\pi}\\approx 190$MeV with $M_{\\pi}L\\approx 3.9$. The PACS collaboration has recently reported results obtained on a $96^4$ lattice with $M_{\\pi}\\approx 147$MeV and $M_{\\pi}L\\approx 6$ \\cite{UkitaLat2015}. Finally, the FERMILAB and MILC collaborations \\cite{Bazavov:2014wgs} have generated a lattice ensemble with $M_{\\pi}\\approx 128$MeV and $M_{\\pi}L\\approx 3.9$.}\n\\begin{table}[tbd]\n\\begin{center}\n\\begin{tabular}{cc|cc|c}\n$M_{\\pi}$ & $\\,M_{\\pi}L\\,$ & $\\,M_{\\pi}\/\\Lambda_{\\chi}\\,$  & $\\,p_n\/\\Lambda_{\\chi}\\,$ & $\\,n_{\\rm max}$ \\\\[0.8ex]\n\\hline\n200 & 4 & $\\simeq 1\/6$ & $\\simeq \\sqrt{n}\/4 $ & 1\\\\\n150 & 4 & $\\simeq 1\/8$ & $\\simeq \\sqrt{n}\/5 $ & 2 \\\\\n130 & 4 & $\\simeq 1\/9$ & $\\simeq \\sqrt{n}\/6 $ & 3 \\\\\n150 & 6 & $\\simeq 1\/8$ & $\\simeq \\sqrt{n}\/8 $ & 5 \\\\\n\\end{tabular}\n\\end{center}\n\\caption{\\label{tab:expansion}  The chiral expansion parameters $M_{\\pi}\/\\Lambda_{\\chi}$ and $p_n\/\\Lambda_{\\chi}$ for various pion masses and spatial extensions $L$. $n_{\\rm max}$ in the last column stems from the condition $p_{n_{\\rm max}}\/\\Lambda_{\\chi}\\approx0.3$}\n\\end{table}\nThe expansion parameter $M_{\\pi}\/\\Lambda_{\\chi}$ is sufficiently small for pion masses smaller than $200$MeV. The situation is less favourable for $p_n\/\\Lambda_{\\chi}$. In order to have at least theoretically a  chiral expansion at all the expansion parameter $p_n\/\\Lambda_{\\chi}$ should be reasonably smaller than 1. If we restrict ourselves to momenta satisfying $p_n\/\\Lambda_{\\chi} \\le 0.3$  we find values for $n_{\\rm max}$ ranging between 2 and 5. For the coefficients $c_n^+$ with $n\\lesssim n_{\\rm max}$ we expect the chiral expansion to be applicable.\n \nBesides the question of applicability there is the question of how rapidly the chiral expansion converges. The smaller $n$ and $p_n$ is the better the chiral expansion behaves.  For $p_n\/\\Lambda_{\\chi} \\simeq0.3$ one does not expect a fast rate of convergence, and the LO results for the coefficients $c_n^{+}$ associated with these rather high momenta will probably receive rather large higher order corrections. Definite statements about these higher order corrections are difficult to make without having done the calculation, but as a rough error estimate we may allow for a 50\\% error. The error will be smaller for the coefficients associated with the smaller momenta. Note that the exponential suppression due to the exponential $\\exp[-(E_{{\\rm tot},n} -M_N) t]$  is stronger for the contributions with larger momentum,  so the contributions with a larger uncertainty in the two-point function are more suppressed.   \n\nAnother reason to constrain the momenta $p_n$ and $n$ stems from the requirement that the nucleon-pion-state energy should be sufficiently well separated form the first one-particle resonance energy. If that is not the case one expects large mixing between these two states that significantly alter the coefficents $c_n^{+}$ \\cite{Luscher:1991cf}.  In practice, however, this seems to be almost automatically satisfied once the momenta are constrained by the bound \\pref{boundpn}, because the energy of the first resonance is about 0.5GeV higher than the nucleon mass. For example, for the values $n_{\\rm max}$  given in table \\ref{tab:expansion} the total energy of the nucleon-pion state is at least 100MeV smaller than the expected resonance energy, and the energy gap for the states with $n<n_{\\rm max}$ is even smaller. We therefore expect mixing effects to be negligible, at least within the uncertainties the LO results presented here are afflicted with.\n\nIn lattice simulations one usually computes the effective nucleon mass, defined as the negative time derivative of $\\log C_+(t)$.  With \\pref{Cplusfinal} we obtain\n\\begin{eqnarray}\nM_{N,{\\rm eff}} & =&  M_N \\left[ 1+ \\sum_{p_n} d^{+}_n e^{-(E_{{\\rm tot},n} -M_N) t}\\right]\\,,\\label{Meff}\\\\ \nd_n^+ & =&  c_n^+\\left[\\frac{E_{{\\rm tot},n}}{M_N}-1\\right]\\,.\n\\end{eqnarray}\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[scale=0.45]{Figures\/plotMeff}\\\\\n\\caption{$M_{N,{\\rm eff}}\/M_N$ as a function of the sink-source separation $t$ for the parameter sets in table \\ref{tab:expansion}. }\n\\label{fig:contMeff}\n\\end{center}\n\\end{figure}\nFigure \\ref{fig:contMeff} shows the ratio $M_{N,{\\rm eff}}\/M_N$ as a function of $t$ for the four parameter sets in table \\ref{tab:expansion}. The deviation from the constant value 1 is caused by nucleon-pion-state contribution. The sum over $p_n$ is truncated at $n_{\\rm max}$ given in table \\ref{tab:expansion}. \nWe conclude that the nucleon-pion-state contribution to the effective mass is small, less than 2 percent for $t\\ge0.5$fm, and dropping below 1 percent at $t\\approx 1$fm. Note that the nucleon-pion-state contribution to the effective mass is significantly smaller than to the two-point function itself since the coefficients $d_n^+$ are smaller than the coefficients $c_n^+$. For the parameter sets of table \\ref{tab:expansion} the factor $E_{{\\rm tot},n}\/M_N-1$ causing this suppression is less than about 0.4. \n\nFigure \\ref{fig:contMeff} shows the results to leading order in the chiral expansion. As argued before we may allow for a 50\\% error for the coefficents, which approximately results in a 50\\% error in figure \\ref{fig:contMeff}. Even with this uncertainty we can conclude that the nucleon-pion-state contribution to the effective mass is at the few-percent level.  Whether this plays a role in practice depends on the statistical errors in the lattice data and the source-sink separations accessible in the simulation. For statistical errors of  1\\% and below the nucleon-pion-state contribution should be noticeable in the data, at least for euclidean times below $0.7$ fm. Obviously, actual lattice data needs to be analyzed with the results presented here before definite conclusions can be drawn.\\footnote{We mention that Ref.\\ \\cite{Mahbub:2013bba} reports on the presence of a non-negligible nucleon-pion-state contribution to $C_-$. Unfortunately, a direct check of our results is not possible.}\n \n\\section{\\label{secConcl} Concluding remarks}\n\nThe results we find for the nucleon-pion-state contribution to the nucleon correlation function is  small. \nWhether it is too small to be seen in lattice QCD data is not easy to answer and depends on various parameters of the simulation, among others on the range for the euclidean time and the size of the statistical errors in the data. Future physical-point simulations aiming at a one-percent error for the nucleon mass will probably be sensitive to the corrections studied here.\n\nWe repeatedly mentioned that the calculations in this paper are LO calculations. In principle, the higher order corrections can be calculated straightforwardly. Whether this is useful in practice is somewhat doubtful. The NLO contributions will depend on the LECs associated with the interpolating nucleon fields. Since their values are a priori unknown the NLO results are in a way less predictive than the LO results derived here. On the other hand, the LO results are universal for all the various nucleon interpolating fields considererd here. In order to discuss and describe differences caused by using different interpolating fields one has to go at least one order higher in the chiral expansion.\n\nHere we studied the dominant nucleon-pion-state  contributions to the nucleon two-point function. The same framework can be used \nto  study the nucleon-pion-state contribution to higher $n$-point functions.  The nucleon axial charge, for example,  is obtained from the three-point function involving the axial vector current. In that case the contamination with excited and multi-particle states is expected to be larger, because more than one euclidean time difference are present in these correlation functions, and these are usually smaller than the source-sink separation in the two-point function. Preliminary results concerning the nucleon-pion-state  contributions in the determination of $g_{\\rm A}$ can be found in \\cite{Bar:2015zha}. Other interesting observables to study are the electromagnetic form factors and the quark momentum fraction in the nucleon, for instance. \n\n\\vspace{2ex}\n\\noindent {\\bf Acknowledgments}\n\\vspace{2ex}\n\nDiscussions with Tomasz Korzec and Stefan Schaefer are gratefully acknowledged. I thank Maarten Golterman and Rainer Sommer for their feedback on a first draft of this manuscript. \n\\vspace{3ex}\n\n\\begin{appendix}\n\n\n\\section{Non-relativistic interpolating fields}\n\\label{appendixA}\n\nAs mentioned at the end of section \\ref{ssect:QCDinterpolaters} non-relativistic interpolating nucleon fields can be constructed from their relativistic counterparts. Here we briefly discuss the minor modification this choice implies in case of $N_1$. It is sufficient to discuss the local interpolating field only, the generalization to the smeared fields is completely analogous to the discussion at the end of section \\ref{ssect:QCDinterpolaters}.\n\nIn our discussion of the relativistic interpolating fields there was no need to specify a particular representation for the $\\gamma$-matrices. For the non-relativistic limit it is useful to assume the Dirac representation with $\\gamma_0$ being diagonal. The reason is that in this basis the projectors $\\Gamma_{\\pm} = (\\gamma_0 \\pm 1)\/2$ can be used to decompose a quark spinor $q$ into its  {\\em upper} (large) spinor component $q_u$ and its {\\em lower} (small) spinor components $q_l$.  The latter vanish in the non-relativistic limit.\n\nDecomposing $N_1$ into upper and lower components one finds that one contribution involves upper components only \\cite{Leinweber:2004it}. Let us call this part $N_{1,{\\rm nr}} = (\\tilde{q}_{u} q_u)q_u$. This field completely breaks charge conjugation but it still transforms according to (2.8) and (2.9) under chiral transformations and parity. So this field is mapped onto the expression \\pref{Neffexp} in the chiral effective theory but the nucleon interpolating field $\\Psi$ in the terms of table \\ref{table:Nop} are replaced by $\\Gamma_+\\Psi$.  Using this  we find the expression \n\\begin{equation}\\label{Nnreffexp}\nN_{1,{\\rm nr}}(x)=  \\tilde{\\alpha}_{\\rm nr} \\left(\\Gamma_+\\Psi(x) -\\Gamma_{\\!-} \\frac{i}{2f} \\pi(x) \\gamma_5\\Psi(x)\\right)\n\\end{equation}\nfor the interpolating field in the chiral effective theory. In addition to a different LEC the parity projectors appear explicitly in this expression, in contrast to the result in \\pref{Neffexp}. \n\nSuppose we compute the positive parity correlation function with this interpolating field. Due to the parity projector $\\Gamma_{+}$ in the definition of the correlator only the first term in \\pref{Nnreffexp} contributes, the second vanishes identically. In terms of Feynman diagrams this means that only diagram e) in fig.\\ \\ref{fig:diagrams} contributes. Dropping the contributions from the other diagrams in the result \\pref{Cnpplus} is very simple, it amounts in dropping the 1 in $[1- g_A \\ldots]^2$ on the right hand side of \\pref{Cnpplus}. This is esily understood since diagram e) is the only one with two vertex insertions and therefore the only one proportional to $g_A^2$. This is the only modification there is, thus the final result for the correlator is the same as in \\pref{Cplusfinal} provided the aforementioned modification is made in the coefficient $h_n^+$.  Note that dropping the 1 in $[1- g_A \\ldots]^2$ makes the coefficient $h_n^+$ larger, even though the increase is not huge, at least for small energies where the contribution proportional to $g_A$ is much larger than 1.\n\nA similarly simple modification is found in the negative parity correlator. In that case only the second term in \\pref{Nnreffexp} provides a non-vanishing contribution to the correlator, thus only diagram b) contributes. This is accounted for by keeping only the 1 in $[1- g_A\\ldots ]^2$ in \\pref{Cnpminus} and \\pref{Cminusfinal}, respectively.\n\n\\end{appendix}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\\label{sec1}\n\nDue to the huge computational overhead of traditional super-resolution, it is difficult to be applied to mobile devices with limited computing capabilities. The main goal of lightweight single-image super-resolution (SISR) is to reconstruct super-resolution (SR) images from the low-resolution (LR) one with fewer parameters and calculations~\\cite{yao2020weighted,li2020s,hou2021coordinate}. In the past ten years, deep learning has made amazing achievements in various computer vision tasks, which also greatly promoted the development of SISR. \n\nRecently, many convolutional neural networks (CNNs) based SISR methods have been proposed~\\cite{3dong2015image,5han2015deep,4dong2016accelerating,tian2020coarse,he2019mrfn}. Compared with the traditional methods, CNN-based SISR methods are more versatile and can reconstruct higher-quality SR images with more texture details. In 2014, Dong \\emph{et al.} introduced the deep learning technology into SISR and proposed the first CNN-based SISR model, named SRCNN~\\cite{3dong2015image}. Although SRCNN has only three convolutional layers, its performance has far surpassed traditional methods and achieved state-of-the-art results at the time. Now, we know that deeper and more complex networks can achieve better performance~\\cite{6lim2017enhanced,1ahn2018fast,7haris2018deep,12zhang2018image,17zhang2018residual,li2018multi,zhang2020unsupervised,li2020mdcn}. However, their parameters and calculations are also huge and are difficult to be used on mobile devices. To address this issue, many lightweight SISR models have also been proposed. For instance, CARN~\\cite{1ahn2018fast} is a lightweight residual network composed of multiple residual connections. ECBSR~\\cite{zhang2021edge} is a lightweight and efficient network whose features are extracted in multiple paths. The purpose of these models is to reduce the complexity of the model and facilitate the application in the real world. The demand for lightweight practical models motivates us to propose the Feature Distillation Interaction Weighted Network (FDIWN). The computational costs of FDIWN are lower than most existing lightweight SISR models, but it is not inferior to them in terms of performance.\n\n\\begin{figure*}[t]\n\t\\centerline{\\includegraphics[width=15.5cm]{Fig\/FDIWN1.png}}\n\t\\caption{The architecture of the proposed Feature Distillation Interaction Weighting Network (FDIWN).}\n\t\\label{Figure 2}\n\\end{figure*}\n\nAs we know, as the depth of the network increases, information will be lost during transmission. Therefore, under the constraints of parameters and calculations, how to prevent information loss, and how to make full use of intermediate features is important. To achieve this, we introduce Wide-residual Distillation Interaction Blocks (WDIB) in the Feature Shuffle Weighted Group (FSWG) for pairwise feature fusion, and then the features are shuffled and weighted. This operation can improve model performance while only increasing a small amount of computational cost since the wide-residual attention weighting units are extensively used in WDIB, including Wide Identical Residual Weighting (WIRW) units and Wide Convolutional Residual Weighting (WCRW) units. WIRW and WCRN allow more features to pass and be activated, thereby increasing the transmission and utilization of the features. Meanwhile, the carefully designed Self-Calibration Fusion (SCF) unit integrates different levels of features by the jump splicing strategy to achieve a good SR reconstruction. In general, our main contributions can be summarized as follows: \n\n\\begin{itemize}\n\\item We propose a wide-residual attention weighting unit for lightweight SISR, including Wide Identical Residual Weighting (WIRW) unit and Wide Convolutional Residual Weighting (WCRW) unit, which has stronger feature distillation capabilities than ordinary residual blocks.\n\\item We propose a novel Self-Calibration Fusion (SCF) module to replace the traditional concatenate operation for efficient feature interaction and fusion, which can aggregate more representative features and self-calibrate the input and output features.\n\\item We propose a Wide-Residual Distillation Connection (WRDC) framework, which connects the coarse and distilled fine features within the module and allows features from different scales to interact with each other. \n\\item We design a Feature Shuffle Weighted Group (FSWG) for pairwise feature fusion, which consists of a series of interactional WDIBs. Meanwhile, it serves as a basic component of the proposed Feature Distillation Interaction Weighting Network (FDIWN).\n\\end{itemize}\n\n\n\\begin{figure*}[t]\n\t\\centerline{\\includegraphics[width=15.5cm]{Fig\/WDIB1.png}}\n\t\\caption{The structure of the proposed Wide-residual Distillation Interaction Block (WDIB). Conv1 and Conv3 represent the convolutional layer with the kernel size of 1 and 3, respectively.}\n\t\\label{Figure 3}\n\\end{figure*}\n\n\\section{Related Work}\n\\label{sec2}\n\n\\subsection{Lightweight SISR}\n\\label{sec21}\n\nRecently, more and more effective deep neural networks have been introduced into SISR. However, most of them are often accompanied by a large number of model parameters and need large calculation costs, which limits their applications on mobile devices. To address this issue, researchers began to explore lightweight and efficient SISR methods. For instance, CARN-M~\\cite{1ahn2018fast} used group convolution to reduce the model parameters, which even achieved better super-resolved effects than some large SISR models. Hui \\emph{et al.}~\\cite{37hui2018fast} presented the Information Distillation Network (IDN) to extract more useful information with fewer convolutional layers. Meanwhile, IMDN~\\cite{11hui2019lightweight} was modified based on the IDN with a faster and lighter structure. After that, RFDN~\\cite{19liu2020residual} changed the channel splitting method based on IMDN and adopted skip connections for the convolutional layers in the residual block. Wang \\emph{et al.}~\\cite{38wang2019lightweight} proposed an adaptive weighted super-resolution network with efficient residual learning and local residual fusion. Wang~\\cite{wang2021lightweight} proposed a multi-scale feature interaction network (MSFIN) for lightweight SISR. IMRN~\\cite{jiang2021learning} uses the pruning method to reduce the model size without significantly reducing the performance, and achieves good performance. Nowadays, lightweight SISR is getting more and more important since its great application value. Although the aforementioned models achieved good results, they ignored the use of intermediate features, resulting in sub-optimal performance.\n\n\n\\subsection{Wide-Residual Attention Weighting Learning}\n\\label{sec22}\n\nStudies have shown that the deeper the network, the better the performance of the model. However, it was later discovered that as the number of the network layers increased, the performance of the model does not rise but falls. To solve this problem, the residual block was introduced into the network, thus the network can reach very deep, and the effect of the network will also become better. This method is also used in many SISR models. For example, VDSR~\\cite{25kim2016accurate} is a 20 layers network, EDSR~\\cite{6lim2017enhanced} is a 65 layers network, and RCAN~\\cite{12zhang2018image} has more than 800 layers. All these models introduced various skip connections and concatenation operations between shallow layers and deep layers to make full use of the shallow feature information. Different from the above methods, Yu \\emph{et al.}~\\cite{27yu2018wide} found that the model with wider features before ReLU activation can achieve better performance. Therefore, they proposed the WDSR with the wide activation mechanism, which expanded features before ReLU and allowed more information to pass through without additional parameters. Meanwhile, the attention mechanism has been widely used in deep learning tasks. For instance, Zhang \\emph{et al.}~\\cite{12zhang2018image} and Dai \\emph{et al.}~\\cite{14dai2019second} proposed the first-order statistics and second-order attention networks to pursue better feature extraction. Inspired by this, we try to introduce the second-order attention mechanism into our modified wide activation residual block to further improve the feature extraction ability of the model.\n\n\\section{Proposed Method}\n\\label{sec3}\nTo build a lightweight and accurate SISR model, we focus on the use of intermediate features and the interaction of feature information. To achieve this, we propose a Feature Distillation Interaction Weighting Network (FDIWN). FDIWN consists of a series of novel and efficient modules, including Wide-residual Distillation Interaction Block (WDIB), Wide-Residual Distillation Connection (WRDC), Wide Identical Residual Weighting (WIRW) unit, Wide Convolutional Residual Weighting (WCRW) unit, and Self-Calibration Fusion (SCF) module.\n\n\\subsection{Feature Distillation Interaction Weighting Network}\n\\label{sec31}\n\nAs shown in Figure~\\ref{Figure 2}, FDIWN consists of three parts: the shallow feature extraction part, the non-linear deep feature acquisition part, and the upsampling recovery part. Following previous works, we use a $3 \\times 3$ convolutional layer to extract the shallow features ${X_0}$ from the input LR image\n\\begin{equation}\n{X_0} = {C_e}({I_{LR}}),\n\\end{equation}\nwhere ${C_e}$ represents the feature extraction layer, ${I_{LR}}$ is the LR image, and ${X_0}$ is the extracted shallow features.\n\nAfter that, the non-linear feature mapping module is followed, which is formed by several Feature Shuffle Weighted Groups (FSWGs) through jump connections. The operation can be expressed as follow\n\\begin{equation}\n{X_1} = F_{FSWG}^0({X_0}),\n\\end{equation}\n\\begin{equation}\n{X_{n - 1}} = F_{FSWG}^{n - 2}( \\ldots (F_{FSWG}^1({X_1}) + {X_1}) \\ldots ) + {X_{n - 2}},\n\\end{equation}\n\\begin{equation}\n{X_n} = F_{FSWG}^{n - 1}({X_{n - 1}}),\n\\end{equation}\nwhere $F_{FSWG}^k$ represents the $k$-th FSWG, and ${X_n}$ denotes the extracted non-linear deep features.\n\nThe features used for the final SR image reconstruction in the upsampling recovery module come from two parts, one comes from the non-linear deep feature extraction module and the other comes from the input LR image. We hope that superimposing the low-frequency and high-frequency feature information in this way can reconstruct high-quality SR images with more texture details. Therefore, the final SR image can be expressed as\n\\begin{equation}\n{I_{SR}} = {F_{UP1}}(\\sum\\limits_{i = 0}^{n - 1} {F_{FSWG}^i({X_i})} )  + {F_{UP2}}({I_{LR}}),\n\\end{equation}\nwhere ${I_{SR}}$ is the reconstructed SR image, ${F_{UP1}}$ and ${F_{UP2}}$ represent the upsampling modules.\n\n\\subsection{Wide-Residual Distillation Connection}\n\\label{sec32}\nAs shown in Figure~\\ref{Figure 3}, Wide-Residual Distillation Connection (WRDC) is an important component in the model, which consists of the Wide Identical Residual Weighting (WIRW) unit, the Wide Convolutional Residual Weighting (WCRW) unit, and the distillation jump connection. Both WIRW and WCRW introduced the wide activation mechanism, thus it can distill richer features with fewer parameters. Recently, with the emphasis on the importance of channel attention in RCAN~\\cite{12zhang2018image}, many SR methods focus on the attention mechanism. As shown in Figure~\\ref{Figure 5}, Zhang et al.~\\cite{31zhang2021sa} proposed a new attention paradigm, which is a combination of spatial attention and channel attention, called Shuffle Attention (SA). Inspired by this, we introduce the SA into our WIRW and WCRW to further enhance their feature extraction abilities. Since the SA mechanism is placed in each wide-residual unit, we set the numbers of the group $g$ to be large enough to keep the SA lightweight. After the channel splitting operation, the number of input channels of WCRW is only half of the original input. Therefore, compared with WIRW, a $3 \\times 3$ convolutional layer is added to the shortcut path of WCRW to increase the number of output channels so that it can match the original input size and achieve the interaction between different features. Meanwhile, WCRW and WIRW both introduce adaptive weights operation in the main path and shortcut path for adaptive feature learning. When input $I$ is fed into WIRW and WCRW, the output ${X_{WIRW}}$ and ${X_{WCRW}}$ can be expressed as\n\\begin{equation}\n{X_{WIRW}} = {\\lambda _{x1}}{F_{SA}}[{F_{CR}}(I)] + {\\lambda _{res1}}I,\n\\end{equation}\n\\begin{equation}\n{X_{WCRW}} = {\\lambda _{x2}}{F_{SA}}[{F_{CR}}(I)] + {\\lambda _{res2}}{F_{C3}}(I),\n\\end{equation}\nwhere ${\\lambda _{xk}}$ and ${\\lambda _{resk}}$ ($k = 1,2$) represent the adaptive multiplier of the $k$-th wide-residual unit branch, ${F_{SA}}$ represents the SA operation, ${F_{CR}}$ represents a series of (conv + relu) operations before the attention mechanism, and ${F_{C3}}$ represents the $3 \\times 3$ convolutional layer. The complete structure of WIRW and WCRW can be seen in Figure~\\ref{Figure 3}. \n\nApart from the above operation, the distillation connection part is applied to segment the channel features through the convolutional layer and Sigmoid function. The convolutional layer is introduced to expand the dimension of the splitting channel, while the Sigmoid function non-linearizes the obtained coarse high-frequency features to obtain fine features maps. Finally, these features are multiplied with the low-frequency attention features obtained after the wide-residual unit refinement process to realize the interaction of the features from different scales.\n\n\\begin{figure}[t]\n\t\\centerline{\\includegraphics[width=8.2cm,trim=0 0 0 0]{Fig\/SA.png}}\n\t\\caption{The principle of shuffle attention (SA) mechanism. $h$, $w$, $c$, and $g$ represent the height, width, number of channels, and number of groups, respectively.}\n\t\\label{Figure 5}\n\\end{figure}\n\n\n\\subsection{Wide-Residual Distillation Interaction Block}\n\\label{sec33}\n\nInspired by the lattice block (LB)~\\cite{32luo2020latticenet}, we design a Wide-Residual Distillation Interaction Block (WDIB) based on WRDC. As shown in Figure~\\ref{Figure 3}, WDIB utilizes the butterfly structure described in LB to realize the interaction of intermediate features. Define ${W_{ir}}$ and ${W_{cr}}$ represent the WIRW and WCRW units, the first butterfly structure can be expressed as\n\\begin{equation}\n{X_{remain1}},{X_{distill1}} = Spli{t_1}({W_{ir}}({X_{in}})),\n\\end{equation}\n\\begin{equation}\n{U_{1}} = {M_1}\\left\\langle {{X_{in}}} \\right\\rangle  + {W_{cr}}({X_{in}}),\n\\end{equation}\n\\begin{equation}\n{V_{1}} = {N_1}\\left\\langle {{W_{cr}}({X_{remain1}})} \\right\\rangle  + {X_{in}},\n\\end{equation}\nwhere $Spli{t_i}(\\cdot)$ represents the $i$-th channel splitting operation, ${X_{remaini}}$ represents the rough feature of the input subsequent wide-residual unit, and ${X_{distilli}}$ represents the $i$-th refinement feature that is split out and jump-connected with the next butterfly structure. The combination coefficients ${M_{i}}$ and ${N_{i}}$ are the two vectors connecting the upper and lower branches. It is worth noting that $M\\left\\langle {X_{in}} \\right\\rangle = M({X_{in}}) \\times {X_{in}}$, and the learning details of the combination coefficients ${M_i}$ and ${N_i}$ are provided in Figure~\\ref{Figure 4}. ${U_{i}}$ and ${V_{i}}$ are the output features after the $i$-th butterfly structure, and then they are fed into the second butterfly structure\n\\begin{equation}\n{X_{remain2}},{X_{distill2}} = Spli{t_2}({W_{ir}}({V_{1}})),\n\\end{equation}\n\\begin{equation}\n{U_2} = {M_2}\\left\\langle {{W_{cr}}({X_{remain2}})} \\right\\rangle  + {U_{1}},\n\\end{equation}\n\\begin{equation}\n{V_2} = {N_2}\\left\\langle {{U_1}} \\right\\rangle  + {W_{cr}}({X_{remain2}}).\n\\end{equation}\n\n\n\\begin{figure}[t]\n\t\\centerline{\\includegraphics[width=8.2cm,trim=0 0 0 0]{Fig\/PC.png}}\n\t\\caption{The diagram of combination coefficient learning.}\n\t\\label{Figure 4}\n\\end{figure}\n\n\nAfter that, the output $X_{out}$ can be expressed as\n\\begin{equation}\n{X_{out1}} = {W_{ir}}({U_2} \\times {S_3}({X_{distill1}})),\n\\end{equation}\n\\begin{equation}\n{X_{out2}} = {W_{ir}}({V_2} \\times {S_3}({X_{distill2}})),\n\\end{equation}\n\\begin{equation}\n{X_{out}} = {C_{SCF}}[{X_{out1}},{X_{out2}}] + {X_{in}},\n\\end{equation}\nwhere ${C_{SCF}}$ represents the proposed Self-Calibration Fusion (SCF) module. ${X_{out1}}$ and ${X_{out2}}$ represent the upper branch and the lower branch entering the SCF module, respectively. In addition, ${S_k}$ represents a $k \\times k$ convolutional layer followed by a cascade of Sigmoid function. The structure of the SCF module is provided in Figure~\\ref{Figure 3}. Obviously, the features of the upper and lower branches are first fused, and then the different scale features from two branches are interacted and fused. The output ${X_{SCF}}$ of the SCF module can be expressed as\n\\begin{equation}\n{X_{concat}} = {C_{Concat}}[{\\lambda _{x3}}{X_{out1}},{\\lambda _{x4}}{X_{out2}}],\n\\end{equation}\n\\begin{equation}\n{X_{SCF}} = {\\lambda _{x4}}{X_{out2}} \\times {S_1}({X_{concat}}) + {W_{cr}}({X_{concat}}),\n\\end{equation}\nwhere ${C_{Concat}}$ represents the Concat operation, ${\\lambda _{x3}}$ and ${\\lambda _{x4}}$ represent the adaptive weights, and ${X_{concat}}$ represents the output after the upper and lower branches are multiplied by the adaptive weight and then the Concat is performed. Subsequently, the fused features are nonlinearized, then multiplied with the features of the lower branch, and finally added to the refined fusion features to achieve the interaction of features from different scales. Since there are a large number of adaptive multipliers in the module, the output features can be adjusted and calibrated continuously during the training, thus it can achieve better performance than the traditional Concat operation.\n\n\n\\subsection{Feature Shuffle Weighted Group}\n\\label{sec34}\n\nAs shown in Figure~\\ref{Figure 2}, Feature Shuffle Weighted Group (FSWG) consists of three interactional WDIBs and it serves as the basic component of FDIWN. Specifically, we fuse and shuffle the features extracted by WDIB one by one. The cascaded operation ${F_{CGS}}$ can be expressed as\n\\begin{equation}\n{F_{CGS}} = {F_{Shuffle}}({F_{GConv}}({C_{Concat}}[{x_i},{x_{i + 1}}])),\n\\end{equation}\nwhere ${F_{Shuffle}}$ represents the channel shuffle operation and ${F_{GConv}}$ represents the group convolution operation. ${x_i}$ and ${x_{i + 1}}$ represent the two features to be merged. After that, we add the shuffled and fused features with the original features that have not been operated to achieve the interaction of feature information. Meanwhile, we set a larger group in the shuffle and fusion operation to reduce the parameter burden. Moreover, to reduce the redundant information, the primary features and the features after the information interaction are self-adaptively fused to distill the desired important features. Define the input of FSWG as ${W_0}$, the output ${W_{out}}$ can be formulated as \n\\begin{equation}\n{W_{CGS}} = {F^2}_{CGS}({F^1}_{CGS}({W_1},{W_2}),{W_3}),\n\\end{equation}\n\\begin{equation}\n{W_{out}} = {\\lambda _x}({W_{CGS}} + {W_3}) + {\\lambda _{res}}{W_{ir}}({W_0}),\n\\end{equation}\nwhere ${W_i}$ represents the output of the $i$-th WDIB, ${F^i}_{CGS}$ represents the $i$-th ${F_{CGS}}$ operation, ${W_{CGS}}$ represents the extracted features after a series of shuffle and fusion operations. In addition, ${\\lambda _x}$ and ${\\lambda _{res}}$ are used to adaptively adjust the weight of each channel. After these operations, the features from each WDIB are adequately interacted and distilled to achieve better SR image reconstruction.\n\n\n\\section{Experiments}\n\\label{sec4}\n\n\\subsection{Datasets and Evaluation Metrics}\n\\label{sec41}\nFollowing previous works, we use the DIV2K~\\cite{agustsson2017ntire} as the training dataset, which contains 800 pairs of images. For testing, we use Set5~\\cite{bevilacqua2012low}, Set14~\\cite{zeyde2010single}, BSDS100~\\cite{martin2001database}, and Urban100~\\cite{huang2015single} to verify the effectiveness of the proposed FDIWN. Meanwhile, two metrics on the Y channel in the YCbCr color space, namely PSNR and SSIM are used to evaluate the model performance.\n\n\\subsection{Implementation Details}\n\\label{sec42}\n\nEach mini-batch during the training consists of 16 RGB image blocks with the size of $48 \\times 48$, which are randomly cropped from the LR image. Meanwhile, the training dataset is enhanced by random and horizontal rotation at different angles for data augmentation. The learning rate is initialized to 2e-4 and a total of 1000 epochs are updated. We implement our model with the PyTorch framework and update it with Adam optimizer. All our experiments are performed on NVIDIA RX 2080TI GPUs. \n\nAs for the model set, the final version of FDIWN consists of 6 FSWGs, while the tiny version of FDIWN-M only consists of 4 FSWGs. The number of input channels is initialized to 24 and the value of the adaptive weight is 1. \n\n\\begin{table}[t]\n \\centering\n  \\setlength\\tabcolsep{3pt}\n  \\renewcommand\\arraystretch{0.4}\n  {\\fontsize{9pt}{12pt}\\selectfont\n  \\begin{tabular}{@{}ccccccc@{}}\n   \\toprule\n    Method  &WR&DC  &SCF &Params  &Multi-adds   & PSNR~~~SSIM        \\\\\n   \\hline\n   Baseline1       &\\XSolid  &\\XSolid   &\\XSolid          &59K    &3.3G    & 37.52~~~0.9587  \\\\\n            Baseline2   &\\Checkmark     &\\XSolid    &\\XSolid            &59K    &4.9G    & 37.53~~~0.9589   \\\\\n   FDIWN        &\\XSolid  &\\XSolid     &\\Checkmark       &89K    &6.5G     & 37.58~~~0.9591     \\\\\n   FDIWN       &\\Checkmark   &\\Checkmark     &\\XSolid      &65K    &6.5G    & 37.59~~~0.9590       \\\\\n   FDIWN       &\\Checkmark   &\\Checkmark  &\\Checkmark        &96K    &9.7G    & \\textbf{37.64}~~~\\textbf{0.9592}  \\\\ \\bottomrule \n  \\end{tabular}}\n  \\caption{Impact analysis of WRDC and SCF.}\n  \\label{tab1}\n\\end{table}\n\n\n\\begin{table}[t]\n\\centering\n  \\setlength\\tabcolsep{3pt}\n  {\\fontsize{9pt}{12pt}\\selectfont\n  \\begin{tabular}{@{}ccccccc@{}}\n   \\toprule\n   Case & Method &Channels &Params   &Multi-adds     & PSNR~~~SSIM   \\\\ \\hline\n   1    & Baseline  & 24    & 152K  & 23.2G     & 37.70~~~0.9594          \\\\\n   2    & FDIWN     & 48  & 96K  & 9.7G     & 37.64~~~0.9592           \\\\\n   3    & FDIWN     &120    &131K  & 9.7G & \\textbf{37.72}~~~\\textbf{0.9596}  \\\\ \\bottomrule \n  \\end{tabular}}\n  \\caption{Impact analysis of WIRW and WCRW.}\n  \\label{tab2}\n\\end{table}\n\n\\subsection{Ablation Studies}\n\\label{sec43}\n\nTo verify the efficiency and effectiveness of the proposed modules, we conducted a series of ablation studies and all of these studies are tested on the Set5 dataset.\n\n\\textbf{The effectiveness of WRDC and SCF.} To verify the effectiveness of WRDC and SCF, we replace the WRDC module in FDIWN with a three-layer cascaded $3 \\times 3$ convolution plus ReLU layer, and replace the SCF module with the concatenate operation. We set this structure as the Baseline1. Baseline2 has a similar structure while the three-layer cascaded convolution plus ReLU layer is placed by the cascaded WIRW plus WCRW units. Then we add WRDC and SCF step by step and compare their performance with the Baseline1. It can be observed from Table~\\ref{tab1} that our proposed FDIWN improves the performance of Baseline1 by 0.12 dB, which proves the effectiveness of the WRDC and SCF module. Specifically, with the DC mechanism, PSNR is increased from 37.53 dB to 37.59 dB, while the number of parameters only increases 6K. Meanwhile, the SCF module can provide a 0.06 dB improvement in model performance with an acceptable increase in the number of parameters.\n\n\\textbf{The effectiveness of WIRW and WCRW.} To evaluate the role of our wide-residual units in the module, we replaced all WIRW and WCRW units in our module with the basic residual block and treat this structure as the baseline model. The kernel size of the convolutional layer in the basic residual block is $3 \\times 3$. To explore the impact of the number of channels before the activation function in our designed wide-residual units on the SR performance, we set the number of channels as 48 and 120, respectively. Due to the lightweight character of the 1$\\times$1 convolution function, we can see from Table~\\ref{tab2} that \\romannumeral 1) Compared with the Baseline model (Case 1 and 3), FDIWN achieves better performance with fewer parameters and computational costs; \\romannumeral 2) Increasing the number of channels (Case 2 and 3), the model performance can be further improved. This fully demonstrates the effectiveness of the introduced wide activation mechanism and the proposed WIRW and WCRW.\n\nTo further verify the impact of WIRW and WCRW units on SISR, we visualize the features produced by the different numbers of WIRW and WCRW units. The network here is composed of one $3 \\times 3$ convolution and several wide-residual units. Figure~\\ref{keshihua} shows that the $3 \\times 3$ convolution can only extract shallow image information but cannot extract image details well. However, it can be seen that as the number of WIRW and WCRW units increases, more edge detail information will be extracted. This also means that as the number of WIRW and WCRW units increases, the ability of the model to capture high-frequency information will be greatly improved. This further demonstrates the effectiveness of WIRW and WCRW.\n\n\\begin{figure}[t]\n\t\\centerline{\\includegraphics[width=8cm,trim=0 0 0 0]{Fig\/keshihua.png}}\n\t\\caption{Feature visualization of different module.}\n\t\\label{keshihua}\n\\end{figure}\n\n\\begin{table}[t]\n\\centering\n  \\setlength\\tabcolsep{4pt}\n  {\\fontsize{9pt}{12pt}\\selectfont\n  \\begin{tabular}{@{}cccccc@{}}\n   \\toprule\n    Method     &BI   &WIRW   &Params     &Multi-adds     & PSNR~~~SSIM        \\\\\n   \\hline\n                                 \\hline\n   Baseline    &\\XSolid    &\\XSolid  &215K    &22.0G                 & 37.81~~~0.9598          \\\\\n   FDIWN       &\\Checkmark   &\\XSolid  &225K    &24.4G               & 37.85~~~\\textbf{0.9600} \\\\\n   FDIWN       &\\Checkmark   &\\Checkmark     &230K        &24.4G               & \\textbf{37.88}~~~\\textbf{0.9600}  \\\\ \\bottomrule \n  \n  \\end{tabular}}\n  \\caption{Evaluation of the combination structure of WDIB. The best results are highlighted. BI: Block Interaction.}\n  \\label{tab3}\n\\end{table}\n\n\n\\begin{figure}[t]\n   \\centering\n   \\includegraphics[width=8.3cm]{Fig\/1.png}\n   \\\\\n   \\includegraphics[width=8cm]{Fig\/PSNR1.pdf}\n   \\caption{Study of different numbers of FSWGs and WDIBs.}\n   \\label{curve}\n\\end{figure}\n\n\\textbf{The combination structure of WDIB.} The FSWG in FDIWN is composed of three WDIBs, which are interacted with each other to yield more representative features. To verify the effect of this combination of WDIB, we chose three cascaded WDIBs as the baseline. In other words, three WDIBs are simply connected without any interaction. According to Table~\\ref{tab3}, we can observe that the information blending structure we used is more effective. It improves the PSNR from 37.81 dB to 37.85 dB with limited computational costs, which further proves the effectiveness of the proposed information interaction structure. In addition, the long-skip connection provided by the WIRW unit can further improve model performance. All these experiments show that the combined structure of WDIB is effective.\n\n\\textbf{Efficiency trade-off.} In Figure~\\ref{curve}, we show the performance change under different numbers of FWSGs and WDIBs. Among them, the circular point $Gm$ denotes that $m$ FSWGs are cascaded. According to the figure, we can observe that \\romannumeral 1) As the number of FSWG and WDIB increases, the model performance can be further improved; \\romannumeral 2) The PSNR does not increase when the number of WDIB increases from 3 to 4. Therefore, we use 6 FSWGs and 3 WDIBs in the final version model to achieve a good balance between model performance, size, and computational costs. \n\n\\begin{figure}[t]\n   \\centering\n   \\includegraphics[width=8cm]{Fig\/urban100x2.png}\n   \\caption{Inference speed study on Urban100 with x2 SR.}\n   \\label{time}\n\\end{figure}\n\n\\begin{figure}[h]\n\t\\centerline{\\includegraphics[width=9cm]{Fig\/Tradeoff_2.png}}\n\t\\caption{Investigations of the model size and performance.}\n\t\\label{size}\n\\end{figure}\n\n\n\n\n\\textbf{Model complexity analysis.} In Figure~\\ref{time}, we show the execution time comparison with other classic lightweight SISR models. Obviously, FDIWN achieves competitive results with fewer parameters. Although our FDIWN is not the fastest model, the execution time is acceptable. Meanwhile, we show the multi-adds comparison between FDIWN and other SISR methods in Figure~\\ref{size}. Obviously, our FDIWN achieves the best balance. Therefore, we can draw a conclusion that FDIWN gains a better trade-off between model size, performance, inference speed, and multi-adds.\n\n\n\\begin{table*}[t]\n \\centering\n  \\setlength\\tabcolsep{1.3mm}\n  {\\fontsize{9pt}{10pt}\\selectfont\n  \\begin{tabular}{@{}l|c|c|c|cccc@{}}\n   \\toprule\n    & &   &   & Set5  &Set14    & BSDS100  & Urban100 \\\\\n   Algorithm           &Scale        &Params &Multi-adds & PSNR~~~SSIM     & PSNR~~~SSIM          & PSNR~~~SSIM    & PSNR~~~SSIM \\\\ \\midrule                                                                                                                            \n           \n   SRCNN~\\cite{3dong2015image}        &   & 57K & 52.7G  & 32.75~~~0.9090        & 29.30~~~0.8215            & 28.41~~~0.7863    & 26.24~~~0.7889     \\\\\n  \n   VDSR~\\cite{25kim2016accurate}      &   & 665K & 612.6G  & 33.67~~~0.9210        & 29.78~~~0.8320          & 28.83~~~0.7990    & 27.14~~~0.8290    \\\\\n   DRCN~\\cite{34kim2016deeply}          &   & 1774K &17974.3G  & 33.82~~~0.9226        & 29.76~~~0.8311            & 28.80~~~0.7963    & 27.15~~~0.8276    \\\\\n   IDN~\\cite{37hui2018fast}          &    & 590K &105.6G  & 34.11~~~0.9253        & 29.99~~~0.8354            & 28.95~~~0.8013    & 27.42~~~0.8359    \\\\\n   CARN-M~\\cite{1ahn2018fast}         &                      & 412K & 46.1G  & 33.99~~~0.9236        & 30.08~~~0.8367            & 28.91~~~0.8000    & 27.55~~~0.8385    \\\\\n   CARN~\\cite{1ahn2018fast}         &                      & 1592K & 118.8G  & 34.29~~~0.9255        & 30.29~~~0.8407            & 29.06~~~0.8034    & 28.06~~~0.8493    \\\\\n   IMDN~\\cite{11hui2019lightweight}         &$ \\times 3$ & 703K  & 71.5G           & 34.36~~~0.9270        & 30.32~~~0.8417                            & 29.09~~~0.8046    & 28.17~~~0.8519    \\\\\n   AWSRN-M~\\cite{38wang2019lightweight}          &                      & 1143K & 116.6G  & 34.42~~~0.9275        & 30.32~~~0.8419   & 29.13~~~0.8059    & 28.26~~~0.8545    \\\\\n   MADNet~\\cite{35lan2020madnet}          &   & 930K & 88.4G           & 34.16~~~0.9253       & 30.21~~~0.8398                             & 28.98~~~0.8023    & 27.77~~~0.8439    \\\\\n   RFDN~\\cite{19liu2020residual}          &   & 541K & 55.4G           & 34.41~~~0.9273        & 30.34~~~0.8420                             & 29.09~~~0.8050    & 28.21~~~0.8525    \\\\\n   MAFFSRN~\\cite{40muqeet2020multi}                  &    & 418K  & 34.2G                & 34.32~~~0.9269        & 30.35~~~0.8429   & 29.09~~~0.8052    & 28.13~~~0.8521    \\\\\n   LAPAR-A~\\cite{36li2021lapar}                             &    & 594K & 114G           & 34.36~~~0.9267        & 30.34~~~0.8421                              & 29.11~~~0.8054   & 28.15~~~0.8523  \\\\\n   \\textbf{FDIWN-M (Ours)}       &   & 446K &35.9 G  & 34.46~~~0.9274     & 30.35~~~0.8423    & 29.10~~~0.8051    & 28.16~~~0.8528   \\\\\n   \\textbf{FDIWN (Ours) }      &   & 645K &51.5G  & \\textbf{34.52}~~~\\textbf{0.9281}   &\\textbf{30.42}~~~\\textbf{0.8438}  &\\textbf{29.14}~~~\\textbf{0.8065}   &\\textbf{28.36}~~~\\textbf{0.8567}                                                                                                       \\\\ \n   \\hline\n    SRCNN~\\cite{3dong2015image}           &    & 57K & 52.7G  & 30.48~~~0.8628        & 27.49~~~0.7503                  & 26.90~~~0.7101     & 24.52~~~0.7221     \\\\\n  \n   VDSR~\\cite{25kim2016accurate}      &    & 665K & 612.6G  & 31.35~~~0.8838        & 28.01~~~0.7674         & 27.29~~~0.7251    & 25.18~~~0.7524    \\\\\n   DRCN~\\cite{34kim2016deeply}          &    & 1774K &17974.3G  & 31.53~~~0.8854        & 28.02~~~0.7670            & 27.23~~~0.7233    & 25.14~~~0.7510    \\\\\n   LapSRN~\\cite{41lai2017deep}                   &    & 813K &149.4G                  & 31.54~~~0.8850        & 28.19~~~0.7720        & 27.32~~~0.7280    & 25.21~~~0.7560    \\\\\n   IDN~\\cite{37hui2018fast}                         &    & 590K &81.9G  & 31.82~~~0.8903        & 28.25~~~0.7730           & 27.41~~~0.7297    & 25.41~~~0.7632    \\\\\n   CARN-M~\\cite{1ahn2018fast}         &                      & 412K & 32.5G  & 31.92~~~0.8903        & 28.42~~~0.7762            & 27.44~~~0.7304    & 25.62~~~0.7694    \\\\\n   CARN~\\cite{1ahn2018fast}         &  & 1592K & 90.9G  & 32.13~~~0.8937        & 28.60~~~0.7806            & 27.58~~~0.7349    & 26.07~~~0.7837    \\\\\n   IMDN~\\cite{11hui2019lightweight}        &$ \\times 4$    & 715K         & 40.9G           & 32.21~~~0.8948        & 28.58~~~0.7811                             & 27.56~~~0.7353   & 26.04~~~0.7838    \\\\\n   AWSRN-M~\\cite{38wang2019lightweight}          &                      & 1254K & 72.0G & 32.21~~~0.8954       & 28.65~~~\\textbf{0.7832}    & 27.60~~~0.7368    & 26.15~~~0.7884    \\\\\n   MADNet~\\cite{35lan2020madnet}        &    & 1002K       & 54.1G           & 31.95~~~0.8917        & 28.44~~~0.7780                            & 27.47~~~0.7327    & 25.76~~~0.7746    \\\\\n   RFDN~\\cite{19liu2020residual}                  &    & 550K         & 31.6G           & \\textbf{32.24}~~~0.8952        & 28.61~~~0.7819     & 27.57~~~0.7360    & 26.11~~~0.7858    \\\\\n   MAFFSRN~\\cite{40muqeet2020multi}                  &    & 441K  & 19.3G                & 32.18~~~0.8948        & 28.58~~~0.7812                            & 27.57~~~0.7361    & 26.04~~~0.7848    \\\\\n   ECBSR~\\cite{zhang2021edge}  &    & 603K  & 34.73G           & 31.92~~~0.8946        & 28.34~~~0.7817                            & 27.48~~~\\textbf{0.7393}    & 25.81~~~0.7773    \\\\\n   LAPAR-A~\\cite{36li2021lapar}                             &    & 659K  & 94G                  & 32.15~~~0.8944       & 28.61~~~0.7818    & 27.61~~~0.7366    & 26.14~~~0.7871  \\\\\n   \\textbf{FDIWN-M (Ours)}              &    & 454K  & 19.6G  & 32.17~~~0.8941       & 28.55~~~0.7806        & 27.58~~~0.7364    &26.02~~~0.7844     \\\\\n   \\textbf{FDIWN (Ours) }              &    & 664K  &28.4G   & 32.23~~~\\textbf{0.8955}   &\\textbf{28.66}~~~0.7829  &\\textbf{27.62}~~~0.7380   &\\textbf{26.28}~~~\\textbf{0.7919}                                                 \\\\ \\bottomrule\n  \\end{tabular}}\n   \\caption{Quantitative comparison with state-of-the-art models. The best results are highlighted.}\n  \\label{tab4}\n\\end{table*}\n\n\\subsection{Comparisons with State-of-the-Art Methods}\n\\label{sec44}\n\nIn Table~\\ref{tab4}, we provide the detail quantitative comparison with several representative SOTA SISR methods. According to the table, we can observe that \\romannumeral 1) Models with a similar number of parameters to our model perform worse than ours; \\romannumeral 2) The models with the same effects have more parameters than ours. Therefore, we can draw a conclusion that our proposed FDIWN-M and FDIWN stand out from these methods and perform very competitively in balancing model size, performance, and computational cost.\n\nFigure~\\ref{Figure 9} allows us to visually compare our method with other advanced methods on the Urban100 dataset. Through a horizontal comparison of the SR results, we can qualitatively see the effectiveness and excellence of our proposed method. Meanwhile, their PSNR and SSIM results are also provided. Our method not only has better visual details but also outperforms existing advanced methods in terms of quantitative data comparisons. \n\n\n\n\\begin{figure*}\n\t\\centerline{\\includegraphics[width=16cm, trim=0 50 0 0]{Fig\/koutu.png}}\n\t\\caption{Visual comparisons on the Urban100 dataset. Due to the page limit, please zoom in for details.}\n\t\\label{Figure 9}\n\\end{figure*}\n\n\n\\section{Conclusions}\n\nIn this paper, we proposed an effective and lightweight Feature Distillation Interaction Weighting Network (FDIWN) for SISR. Compared to other lightweight SISR models, FDIWN not only reduces the computational overhead but also improves the SR performance. In summary, the improvement of our FDIWN is mainly due to the following parts: (\\romannumeral 1) The specially designed wide-residual weighting units (including WIRW and WCRW) have a stronger ability to distill useful features than ordinary residual blocks; (\\romannumeral 2) The shuffle attention (SA) mechanism makes the feature extraction concentrating on the key information; (\\romannumeral 3) The well-designed wide-residual units based WRDC module and SCF module can flexibly aggregate and distill more representative features, allowing features from different scales to efficiently interact with each other. Therefore, the contextual and intermediate features can be well interacted, which benefits high-quality SR image reconstruction. Evaluation results on benchmarks have shown that the proposed FDIWN achieved a good balance between model size, performance, and computational cost.\n\n\\section{Acknowledgments}\nThis work was supported in part by the National Natural Science Foundation of China (Nos. 61972212, 61772568, 62076139 and 61833011), the Natural Science Foundation of Jiangsu Province (No. BK20190089), and the Six Talent Peaks Project in Jiangsu Province (No. RJFW-011).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nNetworks having very high degree of interconnection are vulnerable to the disruptive events such as floods, earthquakes, power outages, electronic attacks, etc. Such regional damages are usually unpredictable and may simultaneously destroy multiple network facilities in a specific geographical area, which result in a long period of network outages. For example, the east Japan earthquake on March 2011 caused 385 telephone offices stopping operation immediately, cut off millions of users from the telephone service and even the emergency restoration took more than one month \\cite{sakano2013disaster}.\n\n\nThe conventional techniques to maintain network continuity can not work well in case of disasters. Network protection, which relies on the expensive pre-allocated backup resources, may fail to deal with regional damage when the backup resources corrupt simultaneously with the primary ones. The restoration mechanism, which computes new routes based on the actual status of network, may introduce too long convergence time to meet the requirements of mission-critical and real-time applications, and leads to serious consequences like transient loops and blackholes\\cite{kompella2007detection}.\n\nTo build disaster-resilient networks, this paper focuses on leveraging the technique of SDN (Software Defined Networking) which provides more intelligent and flexible network management.\n\n SDN networks, such as OpenFlow - enabled \\cite{mckeown2008openflow} networks, decouple the network control plane from the data plane, and have been successfully deployed in the operator's WAN and corporation's LAN to provide robust network services, e.g., the global carrier NTT communications networks, Google and Microsoft inter-datacenter WANs, etc \\cite{jain2013b4,hong2013achieving}. Due to its intrinsic great flexibility and global management of the network, SDN is potentially suitable to execute an efficient recovery during a major disruption.\n\nAlthough SDN has a good potential for handling failures, the current architecture may be not sufficient to recover from large-scale failures such as disaster failures.\n\\emph{Control plane scalability} and the \\emph{recovery time requirement} are the two major challenges.\nGenerally, the SDN controller computes routes whenever a failure occurs. And the controller is responsible for updating all the forwarding elements' status. However, the multiple failures caused by a catastrophic event will simultaneously disrupt a lot of end nodes. This will lead to a huge amount of reconnection requests, making it impractical to offload the task of all the routing computation and to update the forwarding elements' status to the controller.\nThis is because the dynamic route re-computation can lead to huge overhead, and inserting all new routes into SDN forwarding elements alongside is time-consuming \\cite{tootoonchian2012controller} and error-prone due to the consistent packet processing problem \\cite{katta2013incremental,peresini2013cpp}.\nMoreover, the stringent recovery time requirements of mission-critical and real-time applications \\cite{liotine2003mission} make the enhancement design of the control plane (e.g., the distributed control plane design like Onix\\cite{koponen2010onix}) incompetent. This is because the status synchronization among physically distributed controllers requires additional time.\n\nIn this paper, we propose a new framework to deal with the considered problems in face of disaster failures by SDN. Several design challenges are addressed in this paper,\n\\begin{enumerate}\n\\item[(1)] \\emph{Low controller overhead}. The controller overhead should be low in order to reduce the likelihood of controller being the bottleneck.\n\\item[(2)] \\emph{Fast recovery}. The recovery should be quick in order to meet the requirement of some mission-critical and real-time applications.\n\\item[(3)] \\emph{Strong connectivity}. The connectivity ratio should be high even after a major disaster event destroying a lot of network components.\n\\end{enumerate}\n\n\nTo address above challenges, our main idea is to pre-install redundant flow entries (backup entries) into the data plane. Different from the previous enhancement design of the control plane, our enhancement design of the data plane guarantees that a large proportion of the reconnection requests can be handled on the data plane. Since only a small fraction of the requests are handled by the controller, the control overhead is low. Besides, the data plane handled requests will not be sent to the control plane, thus saving the round-trip recovery delay between the data plane and the control plane. To address the third challenge, we consider the disaster failure's geographical layout and its failure size distribution. In order to do this, we adopt the novel metric of the vulnerable zone of a path during generating the backup entries. So the pre-installed backup routes are less likely to simultaneously get destroyed by a disaster failure. By combing with the recovery on the control plane, our design guarantees strong connectivity after a disaster failure.\n\nMore concretely, our proposed design consists of two modules: the proactive local failure recovery module running on the switches (data plane) and the reactive global restoration module running on the controller (control plane). In the protection module, we adopt the multi-topology routing to do local fast rerouting, and consider the geography properties (shape and size) of the disaster failure to generate robust backup routes. In the restoration module, we give an effective algorithm to reconnect failed nodes by rescheduling the pre-installed routes. We further consider the load balance performance during recovery by formulating an ILP, after which an heuristic algorithm is proposed. We implement the prototype by utilizing multiple tables pipeline processing and fast failover group tables of OpenFlow (Section \\ref{sec:system_design}).\nSimulations ( Section \\ref{sec:performance_evaluation}) on both random generated and realistic topologies show that, the protection module is able to handle approximately 70\\% of the reconnection requests. The rests are processed by the restoration module. Only by rescheduling the pre-installed redundancies, more than 90\\% of the disconnected end nodes can be reconnected even when the failure diameter is $1\/6$ of the network deployment region's length.\n\\section{preliminary}\\label{sec:preliminary}\nIn this section, we first introduce the network model and failure mode adopted in this paper. Then we introduce the vulnerable zone of a routing path. \\subsection{Network Model}\nWe consider a physical network $G(V, E)$ as a planar graph inside the deployment area $D\\in \\mathbb{R}^{2}$, which is represented by the network components: $V$ is the set of forwarding elements (routers or SDN switches) and $E$ is the set of links connecting them. In SDN context, all forwarding elements have a channel connected to the central \\emph{controller} $C$ (in-band or out-of-band)\\footnote{The logical controller \\emph{C} can be implemented distributedly\\cite{heller2012controller,nguyen2013software}, this refers to the controller placement problem and is out of the scope of our works.}.\nBy $e_{ij}$ we denote the link between adjacent nodes $i$ and $j$, $i,j\\in V, e_{ij}\\in E$. By $x_{st}$ we denote the path between nodes $s$ and $t$, $s,t\\in V$.\n\n\n\n\n\n\\subsection{Failure Model}\nDuring the extreme events such as disasters or malicious attacks, multiple network components located closely to each other may fail together. We summarize the behaviors of such large scale attacks to model the ``the geographically correlated failure''.\n\n\n\\begin{definition} (\\textbf{Geographically Correlated Failure}) is defined as follows:\n\\begin{enumerate}\n\\item Network components intersecting the region of failures will be removed from the network. The size of a geographically correlated failure is determined by the radius $r$.\n\\item The radius $r$ follows the distribution functions $f(r)$, $r_{a}\\leq r\\leq r_{b}$, where $r_{a}$ (resp. $r_{b}$) is the minimum (resp. maximum) considered region size\\footnote{The distribution function $f(r)$ of the destructive natural regional failures, such as earthquakes, usually follows the power-law distribution\\cite{27627372}.}.\n\\end{enumerate}\n\\end{definition}\n\nIt's notable that our model does not make any assumptions about the failure locations and radiuses, which are usually difficult to obtain due to the uncertainty of the disaster failures. Our model is more general than the previous deterministic failure model \\cite{S09rbc, N08ati} (which requires the knowledge of the failure radiuses) and SRLG related model \\cite{SRLG,H03dri, D04drf, S05srl, W10oao, L10dri} (which requires the knowledge of the failure locations).\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=3.0 in]{caochang.eps}\n\t\\caption{Vulnerable zone: the union of points that are located no more than $r$ distance from the network components. Any region failure occurs in the vulnerable zone will break the network component.}\n\t\\label{fig:caochang}\n\\end{figure}\n\\subsection{Vulnerable Zone of a Path}\nAccording to the definition of regional failures, a disaster region can be of any shape with arbitrary size and located anywhere in the plane. Therefore, there are infinite number of region failures to be considered. Our first problem is to find a proper statistical metric to evaluate the impact of region failures.\n\nGiven a regional failure with radius $r$, a link $e_{ij}$ may fail if it intersects with the failure region. In other words, if a disaster happens and its epicenter is less than $r$ distance from $e_{ij}$, $e_{ij}$ will be broken. We call the set of those points the \\emph{vulnerable zone} of link $e_{ij}$, denoted by \\textbf{$\\bm{Z^{r}_{e_{ij}}}$}, defined as follows:\n\\begin{definition} (\\textbf{Vulnerable zone of a link}) is the region sub-area such that any region failure with radius $r$  whose epicenter falls within it will always cause the corruption of the given link.\n\\end{definition}\nAs illustrated in Fig. \\ref{fig:caochang} (a), the ``hippodrome'' in dash line represents the vulnerable zone of link $e_{ij}$, which consists of all points whose shortest distance to link $e_{ij}$ is no more than $r$.\nSimilarly, we can further define the vulnerable zone of a path $x_{st}$, denoted as $\\bm{Z^{r}_{x_{st}}}$ , as shown in Fig. \\ref{fig:caochang} (b), which is the union of all circle centers that are located no more than $r$ distance from the path.\n\\begin{definition} (\\textbf{Vulnerable zone of a path}) is the region sub-area such that any region failure with radius $r$ having epicenter falling within it will always cause the corruption of the given path.\n\\end{definition}\nThe vulnerable zone of a path $x_{st}$ is the union of the vulnerable zones of all the links of the path, i.e., $Z^{r}_{x_{st}}=\\cup_{e_{ij}\\in x_{st}}Z^{r}_{e_{ij}}$.\n\n\\section{system design}\\label{sec:system_design}\nIn this section, we first define the problem of SDN network reliability against regional damage. Then we introduce our system, which consists of two modules: the proactive Backup Topologies Generation Module for local recovery and the reactive Splicing Module for global restoration.\n\n\\subsection{Overview of System Design}\\label{sec:overview-of-system-design}\nThe problem to be solved can be defined as follows: \\emph{Given a network $G(V,E)$ and a central controller $C$, 1) how do we pre-install some redundancies into the network so that the controller is able to reschedule these reduncancies and 2) how does the controller reschedule the protection resources with low controller overhead to survive from the large-scale multiple failures caused by regional damage.}\n\nTo solve the above problem, we apply the SDN framework for failure recovery. Our design consists of two modules, a proactive local failure recovery module working in the forwarding plane (Backup Topologies Generation Module) and a reactive global restoration module running in the control plane (Splicing Module). The failure reconnection requests first get handled by the local failure recovery module. The reconnection requests that the local recovery module is not able to handle are led to the global restoration module as illustrated in Fig. \\ref{fig:system_workflow}.\n\nHow to install the redundancies for the proactive local failure recovery module needs to be carefully addressed. To reduce the controller overhead, we want to handle failures locally as much as possible. However,  the limited number of redundancies on the data plane can not handle all of the failures. We thus have to distinguish the types of failures, so as to decide which of them to handle on the data plane and which on the control plane. This ``distinction'' function can only be implemented on the data plane. Otherwise it would require the interference of the controller which may lead to some additional controller overhead (to decide the types) and additional recovery delay (the round trip time between the control plane and the data plane). To make the distinction on the data plane possible, we refer to the approach of Multi-Topology (MT) Routing (RFC 4915 and RFC 5120 \\cite{psenak2007multi, przygienda2008m}) to add redundancies. And by the joint design of multi-topology redundancies and the multi-table pipeline processing of OpenFlow, this distinction function is made possible without any interference of the controller.\n\nThe basic idea of MT routing is to take the original graph $G$ as input, and generate $k$ backup topologies $\\{G_1 \\ldots G_k\\}$. Routing tables $\\{T_1 \\ldots T_k\\}$ are computed and installed based on $\\{G_1 \\ldots G_k\\}$. Moreover, we notice that the recent research on MT Routing, Multiple Routing Configurations (MRC) \\cite{kvalbein2009multiple,kvalbein2006fast}, is a good technique. The design goal of MRC is to prepare different configurations for different single node or link failures to achieve fast rerouting. With the adoption of MRC, the goal of the distinction function is clear, i.e., to distinguish the single link or node failure and the multiple failure. Furthermore, during the route planning in each backup topology, we consider the geographical distribution of network components to reduce the likelihood of route corruptions by regional failures.\n\nFor the reactive global recovery module running on the control plane to handle the remaining failures, a straightforward idea is to compute new routes on the controller for each failed flow and install all the new rules into the corresponding switches (\\cite{staessens2011software,sharma2012openflow,nguyen2013software}). However, such an operation is time consuming and error-prone due to the consistent packet processing problem\\cite{katta2013incremental,peresini2013cpp}. Instead, we exploit the usage of pre-computed backup topologies to rebuild the failed connections, and a splicing algorithm is proposed in the Splicing Module at the controller to find new paths.\n\n\\begin{figure}[!h]\n\\begin{center}\n\t\t\\includegraphics[width=3.0in]{system-workflow.eps}\n\t\\caption{System overview}\n\t\\label{fig:system_workflow}\n\\end{center}\n\\end{figure}\n\\subsection{Review of MRC (Multiple Routing Configurations)}\nFor the completeness of our work, we first give a brief review of the MRC algorithm. The key idea of MRC routing algorithm is to prepare multiple backup topologies $\\{G_1 \\ldots G_k\\}$, and select a proper backup topology in accordance with the current network failure state \\cite{kvalbein2006fast}. In each backup topology $G_i$, some links $e_{u_{i}v_{i}}$ are defined as \\emph{isolated links} and  \\emph{restricted links} while some nodes $u_i$ are defined as \\emph{isolated nodes}. The isolated links are set infinite weight and can be excluded from $G_i$. The restricted links are set very high weight so that they will not be chosen by some routing algorithms (i.e., shortest path routing mechanisms) unless have to. A node $u_i$ in $G_i$ is isolated if and only if its adjacent links are all either restricted or isolated.  Whenever the isolated nodes fail, it will not affect the connection of other paths\n\nIf node $u$ detects a failure of adjacent link $e_{uv}$, $u$ will select a backup topology $G_i$ in which the failed next hop link $e_{uv}$ is isolated. Then it will tag packets with the selected backup topology id $i$ to notify the subsequent node to forward packets based on this backup topology. Since in $G_i$,  $e_{u_{i}v_{i}}$ is assigned a very high weight and it does not undertake any transit traffic, the packets are guaranteed to reach their destination.\n\nGenerally, to restore an arbitrary single link or node failure, the following constraints must be satisfied:\n\n\\begin{enumerate\n\\item[(1)] Each node $u$ in the original graph must be isolated in at least one backup topology $G_i$. Each link $e_{uv}$ in the original graph must be isolated in at least one backup topology $G_i$.\n\\item[(2)] Each link must be isolated with one of its adjacent isolated nodes in one backup topology.\n\\item[(3)] All node pairs must be mutually reachable in $G_i$.\n\\end{enumerate}\n\n\\begin{figure}[!h]\n\\begin{center}\n\t\t\\includegraphics[scale=0.6]{example-backup-topologies.eps}\n\t\\caption{(a):Original network $G$.(b)-(c):backup topologies $G_1$-$G_3$. Dark node refers to the isolated node and dashed line refers to the restricted link. For clarity, we did not draw the isolated links in $G_i$.}\n\t\\label{fig:example_backup_topologies}\n\\end{center}\n\\end{figure}\n\nFig. \\ref{fig:example_backup_topologies} shows the generated backup topologies. Every node is isolated in exactly one backup topology. Consider a flow with (src=1,dst=3): normally its path is $1\\rightarrow2\\rightarrow3$ based on shortest path in $G$. Assume node 2 failed and node 1 detected the failure. Since node 2 is isolated in $G_1$, node 1 would tag the packet with tag 1 which refers to backup topology $G_1$. The alongside nodes will also forward the packet belonging to the failed flow based on $G_1$. Thus, the routing path from 1 to 3 becomes $1\\rightarrow4\\rightarrow7\\rightarrow8\\rightarrow3$.\n\nThe MRC is originally designed for locally handling single link or node failure. It is not adequate to handle multiple failures caused by large-scale regional failure. To leverage its redundancy, we first modify it based on the geographical distribution of failures to better accommodate the Splicing Module as described in the next subsection.\n\\subsection{Backup Topologies Generation Module}\nThis section first introduce how to generate routes on these backup graphs $\\{G_1 \\ldots G_k\\}$ obtained by the MRC algorithm. Even with a sophisticated path finding algorithm, it's impossible for the limited number of redundancies on the data plane to handle all of the failures. So this module is also responsible for distinguishing the failure types so as to deliver some failures to the control plane to get handled.\n\\subsubsection{Geography based Backup Route Generation}\nAfter running the MRC backup topology generation algorithm, we obtain multiple backup topologies $\\{G_1 \\ldots G_k\\}$. If we run the same path finding algorithm (i.e., Dijkstra algorithm) on all these backup topologies, for specific nodes $s$ and $t$, the path between them, $x_{st}^{i}$ (on $G_i$) and $y_{st}^{j}$ (on $G_j$) , will be possibly coincided. This should be avoided because paths on these backup graphs are too close to each other and are vulnerable to a common risk. To improve the reliability of these paths,\nwe adopt the vulnerable area of  a path during the route generation in each $G_i$.\n\nFor each $s, t\\in G$, we may have different paths on each $G_i$. Compared to the original path in $G$, these paths are redundant. They can be used to rebuild the failed connection between $s$ and $t$.\nHowever, if the vulnerable areas of backup paths intersect, they can be possibly destroyed by a regional failure simultaneously.\nConsider a flow with (src=6, dst=3): normally its primary path in $G$ is $6\\rightarrow 7\\rightarrow 5\\rightarrow 3$ based on the shortest path. The backup path from $6$ to $3$ in $G_1$ is $6\\rightarrow 7\\rightarrow 8\\rightarrow 3$. The backup path in $G_2$ is $6\\rightarrow 4\\rightarrow 5\\rightarrow 3$. The backup path in $G_3$ is $6\\rightarrow 7\\rightarrow 5\\rightarrow 3$. Assume that a regional failure destroys node 5 and node 8 simultaneously. All the primary and backup paths are destroyed. This is because in this case, the primary path and the three backup paths are not region-disjoint. If, however, in $G_3$, we select the region-disjoint path $6\\rightarrow 1\\rightarrow 2\\rightarrow 3$ from $6\\rightarrow 7\\rightarrow 5\\rightarrow 3$ in $G_1$, backup path in $G_3$ would not get destroyed by the regional failure.\n\n\n\nTo avoid the situation in which all the primary path and backup paths are destroyed, it is required that these paths are region-disjoint\\cite{T13cra}. However, finding region-disjoint paths is NP-hard even with a fixed failure radius $r$ \\cite{T13cra} and is difficult to solve in general. Therefore, we refer to heuristic algorithms.\nOur algorithm is shown in Algorithm \\ref{algo:b_t_g}. It first finds the shortest path $x_{st}^{0}$ from $s$ to $t$ on $G$ ($G_0$) as the primary path between $s$ and $t$. Then it iterates on all the backup topologies. $[r_a, r_b]$ is evenly divided into $k$ intervals. Each backup topology $G_i$ is resilient to failures with radius up to $r_{i}=r_a+(i-1)\\cdot\\frac{r_b-r_a}{k-1}$. This is achieved by reducing the likelihood of the vulnerable zone of backup path, $Z_{e_{uv}}^{r_{i}}$ intersecting with the vulnerable zone of the primary path, $Z_{x_{st}^{0}}^{r_{i}}$.\nIf the two vulnerable zones intersect, it means that they can be both destroyed by a failure with radius $r_i$. We assign very high weight to those links whose vulnerable zones intersect with $Z_{x_{st}^{0}}^{r_{i}}$ to reduce the likelihood of choosing those links in $y_{st}^{i}$.\n\n\\begin{algorithm}[!ht]\n\\caption{Backup Routes Generation}\n\\label{algo:b_t_g}\n\\KwIn{network topology $G=(V, E)$, backup topologies $\\{G_1 \\ldots G_k\\}$, source address $s$, destination address $t$}\n\\KwOut{$k$ backup routes in $\\{G_1 \\ldots G_k\\}$}\n\\Begin{\nFind $x_{st}^{0}$ in the original topology $G$\\\\\n\\For{i$\\leftarrow$1 \\KwTo k}{\n$r_{i}:=r_a+(i-1)\\cdot\\frac{r_b-r_a}{k-1}$\\\\\n\\ForAll{edge $e_{uv}\\in E_i$}{\n\\lIf{$Z_{e_{uv}}^{r_{i}}\\cap Z_{x_{st}^{0}}^{r_{i}}\\neq \\emptyset$}{\n    $w_{e_{uv}}^{i}:= very\\ high\\ weight$\n}\n}\nFind $y_{st}^{i}$ in backup topology $G_i$ using the new weight\n}\n\n\\Return $\\{y^{1}_{st} \\ldots y^{k}_{st}\\}$\n}\n\\end{algorithm}\n\n\n\\subsubsection{Implementation}\nIf the packet can not be handled on the data plane, they are sent to the controller to get handled on the control plane. The data plane should be able to deliver the failures to the control plane immediately after finding itself unable to handle them. Unlike the traditional router, the data plane and the control plane in SDN are usually physically separated. Also, the controller should not interfere with this logic. Otherwise, it would prolong the recovery time (due to the round trip time between switches and the controller) and increase the controller overhead.\nWe achieve this by carefully arranging the routes in the pipeline and leveraging the fast failover group table provided in the OpenFlow.\n\nGenerally, the OpenFlow pipeline processing consists of multiple routing tables  $\\{T_0 \\ldots T_{max}\\}$ and a group table $T_g$.\nFlow entries both in $T_i\\ (0\\leq i\\leq max)$ and $T_g$ consist of a lot of terms. In $T_i$, entries consist of match fields, instructions and priority. The failover group entries in $T_g$, consist of group id and action buckets. Each action bucket is associated with a specific port (watch port) that controls the bucket's liveness. The action buckets within an entry are evaluated sequentially. The first bucket which is associated with a live port is selected.\n\nThe conventional routing procedure is to directly forward a packet $p$ to a specific port. In contrast, to leverage the fast failover group, we first forward $p$ to a specific group in the group table. Then it's up to the group to decide which port to forward to based on the port's liveness. The packets are basically divided into two types, i.e., clean packets and dirty packets. The clean packets are those packets that have not encounter any failure yet via routing. The dirty packets have encountered failures before. The clean packets and the dirty packets are processed by different processing flows in the multiple table pipeline. The two types of packets are explicitly distinguished by the MPLS tag in the packet header.\n\nThe detailed procedure is shown in Fig. \\ref{fig:proposed_architecture}. Concretely, $T_0$ is the starting table for all packets, which works as a diverting table to divert packets to different processing flows.\n\\begin{enumerate}\n\\item[(1)] A clean packet $p$ is diverted by $T_0$ to one of the groups in $T_g$. The group which $T_0$ forwards $p$ to, is responsible for checking the liveness of the output port which is based on routing table $T_0$. If the output port is alive, $p$ will be sent out via that output port. Otherwise, it will be tagged a MPLS label $\\o$ ($\\o$ is a backup topology's number) to indicate that it is a dirty packet. Then it will be sent out via another port which is decided by the routing in $T_{\\o}$.\n\\item[(2)] A dirty packet $p$ with a MPLS label $i$ is diverted by $T_0$ to $T_{i}$. Then $T_{i}$ will forward $p$ to one of the groups in $T_g$. The group will check the liveness of the output port based on routing table $T_{i}$. If the port is alive, $p$ will be sent out, otherwise it will be sent to the controller to get further processed.\n\\end{enumerate}\n\n\n\n\\subsection{Splicing Module}\\label{sec:splicing-module}\nThe splicing module refers to the \\emph{reactive} recovery at the controller. It's responsible for rebuilding the failed connections that can't not get handled by the pre-installed redundancies. As described in Section \\ref{sec:overview-of-system-design}, unlike conventional approaches that install all routes into forwarding elements, we rebuild the failed connections by utilizing the pre-installed redundancies. By doing so, the number of installed routes is reduced, thus reducing the likelihood of the consistent packet processing problem. A motivation example is shown in Fig. \\ref{fig:splicing-motivation}. The are two paths from $s$ to $t$, $p1$ and $p2$. A regional failure destroy $e_{ae}$ and $e_{eb}$ simultaneously. As a result, both $p1$ and $p2$ are destroyed. To rebuild the connection, the traditional approach is to install new routes along $s\\rightarrow c\\rightarrow e\\rightarrow d\\rightarrow t$, which requires installing five new rules. In contrast, we only install two rules, one on node $s$ to divert traffic to $p2$, the other one on node $e$ to divert traffic from $p2$ to $p1$.\n\\begin{figure}\n\\centering\n\t\t\\includegraphics[width=2.0 in]{splicing-motivation.eps}\n\t\\caption{A regional failure destroys both $p1$ and $p2$. We rebuild the path by installing one route on node $e$ to divert traffic from $p1$ to $p2$.}\n\t\\label{fig:splicing-motivation}\n\\end{figure}\n\nAnother issue we consider is the load balancing during recovery. Unlike the single link or node failure, the regional failures usually destroy a huge amount of network components simultaneously and lead to a huge amount of disconnected end nodes. Such a huge number of disconnected end nodes requires lots of reconnections. The splicing module should handle the reconnections in a proper way to avoid that some nodes bear exceedingly more rerouting paths than others. The requests that sent to the controller give us the opportunity to redistribute some of the reconnecting traffic. Aside from the connectivity, we also consider the problem of how to reconnect the failed paths. We first define a metric, then we give an ILP to formulate the problem, after which we propose an efficient heuristic algorithm to reduce the complexity.\n\nWe define the metric, maximal load to quantify the routing load balance degree after a regional failure.\n\\begin{definition} (\\textbf{Maximal Load}): given a network graph $G^{'} (V^{'}, E^{'})$ after a regional failure (with failure radius $r$), and a reconnection request matrix $\\mathcal{RM}$ on $G^{'}$, the maximal load gap is the load of the most loaded node in the network, where the load of a node is the number of primary and rerouted paths passing it, i.e.,\n\\begin{align} \\nonumber\nML = |(P_{u}+R^{'}_{u})|_{max}, \\forall u \\in V^{'}\n\\end{align}\n\\end{definition}\nBy $P_{u}$ we denote the number of primary paths that pass  $u$. By $R^{'}_{u}$ we denote the number of the rerouted paths that pass $u$ based on $\\mathcal{RM}$ after a regional failure. Here, we consider the primary resource and the rerouted resource separately, and we assume that if a primary path for a particular node pair is not failed, the primary path can not be altered to avoid network wide reconfiguration. Our goal is to minimize the maximal load.\n\nWe formulate the problem by the following ILP,\n\\begin{alignat}{1}\n\\min \\max_{ i \\in V^{'}}\\quad &\\bigg(\\sum_{\\forall s, t \\in V}\\sum_{e_{ij}\\in E}x^{st}_{ij} + \\sum_{\\forall s, t \\in \\mathcal{RM}_{1}}\\sum_{e_{ij}\\in E^{'}}y^{st}_{ij}\\nonumber \\\\ \\tag{ILP1}\n\\!\\!\\!\\!\\!\\!\\!  & + \\sum_{\\forall s, t \\in \\nonumber\\mathcal{RM}_{2}}\\sum_{e_{ij}\\in E^{'}}z^{st}_{ij}  \\bigg)\\\\ \\nonumber\n\\mbox{s.t.}\\quad\n&z_{st}\\ is\\ a\\ routing\\ path\\tag{1a}\\nonumber\\\\\n&\\!\\!\\!\\!\\!\\!\\!\\! y_{ij}^{st} \\in \\{0, 1\\}, \\forall e_{ij}\\in E,\\forall s, t \\in V^{'} \\tag{1b} \\nonumber\n\\end{alignat}\n\nThe first part in the object function corresponds to $P_{i}$, in which $x_{ij}^{st}$ equals to 1 if path $x_{st}$ passes $e_{ij}$ and 0 otherwise. $x_{ij}^{st}$ is computed based on the routing in $G_{0}$. The second part in the object function corresponds to the additional rerouting paths that $i$ have to undertake due to the reconnection requests $\\mathcal{RM}_{1}$. $\\mathcal{RM}_{1}$ is the reconnection requests that can be handled on the data plane. The second part can also be computed. The third part in the object function computes the number of rerouting paths that $i$ have to undertake due to the reconnection requests $\\mathcal{RM}_{2}$. $\\mathcal{RM}_{2}$ is the reconnection requests that can not be handled on the data plane. It is notable that $\\mathcal{RM}$ equals to the sum of $\\mathcal{RM}_{1}$ and $\\mathcal{RM}_{2}$.\n\nILP1 distribute the rerouting traffic in a min-max fashion. However, the above optimization may not scale well to large networks. In addition to the optimization, we also give a heuristic algorithm. To evenly distribute the reconnections request using the installed redundancies, for all node $u\\in E$, we record $R^{'}_{u}$ on the control plane. We first construct a temporary graph in which, we fill all the available segments (not broken by the regional failure) from $\\{G_1 \\ldots G_{k}\\}$ into the temporary graph. Then weight assignment is performed based on $R^{'}_{u}$ for all node $u\\in E$. The detailed algorithm is in Algorithm 2.\n\nIn Algorithm 2, we first test if $s$ and $t$ are physically disconnected. If they are, it's impossible to find a path between them. Then we construct a multigraph $G_{temp}$ by adding edges from $k$ paths from $s$ to $t$ in $\\{G_1 \\ldots G_k\\}$ excluding the failed edges. To evenly distribute the rerouting paths, we set link weight of $e_{uv}$ to the mean of $R^{'}_{u}$ and $R^{'}_{v}$. After the weight is set, we try to find a path on this temporary graph. If a path is found, we update $R^{'}_{u}, u\\in V$ and return the splicing actions. Otherwise, it means that the failed path can not be rebuild by splicing the existing redundancies. In this case, one may have to install a whole new path.\n\\begin{algorithm}[!ht]\n\\caption{Splicing Action Generation}\n\\label{algo:s_a_g}\n\\KwIn{network topology $G=(V, E)$, backup topologies $\\{G_1 \\ldots G_k\\}$, source address $s$, destination address $t$}\n\\KwOut{a set of splicing actions.}\n\\Begin{\n\\eIf{$s$ and $t$ are physically disconnected}{\n\\Return failed and abort\n}\n{\nbuild a temporal topology $G_{temp}(V_{temp}, E_{temp})$\\\\\n$V_{temp}:=V, E_{temp}:=\\emptyset$\\\\\n\\For{i$\\leftarrow$1 \\KwTo k}{\n\\ForAll{edge $e_{uv}\\in E_i$}{\n\\eIf{$e_{uv}$ is alive and on the path from $s$ to $t$ in $G_i$}{\n$E_{temp}:=E_{temp}\\cup e_{uv}$\\\\\n$\\ell_{e_{uv}}:=i$\n}{\ncontinue\n}\n}\n}\n\\ForAll {edge $e_{uv}\\in E_{temp}$}{\n$w_{e_{uv}}:=(R^{'}_{u}+R^{'}_{v})\/2$\n}\nfind shortest path $p_{temp}$ on $G_{temp}$ from $s$ to $t$\\\\\n\\eIf{found}{\n\\ForAll {edge $e_{uv}\\in E_{temp}$}{\n\\lIf{$e_{uv}\\in p_{temp}$}{\\\\\n$R^{'}_{u}:=R^{'}_{u}+1$\\\\\n$R^{'}_{v}:=R^{'}_{v}+1$\n}\n{}\n}\n}{\\Return failed and abort}\n\\Return splicing actions based on $p_{temp}$\n}\n}\n\\end{algorithm}\n\n\n\\begin{figure}[!h]\n \\begin{center}\n\t\t\\includegraphics[scale=0.80]{pipeline-table-design.eps}\n\t\\caption{Prototype architecture}\n\t\\label{fig:proposed_architecture}\n \\end{center}\n\\end{figure}\n\n\n\\section{performance evaluation}\\label{sec:performance_evaluation}\n\\subsection{Simulation Setting}\n We use both random and realistic topologies for our simulation. The random50 topology contains 50 nodes and 120 edges, the random100 topology contains 100 nodes and 211 edges. The realistic Germany backbone consists of 50 nodes and 88 edges. The deployment area is 1200 x 1200 (arbitrary units) for all the cases. One connection is requested by each node pair of the network. We implement our prototype using OpenFlow 1.3.3\\cite{specification2013version} and NOX controller\\cite{gude2008nox} and examine the prototype's performance on Mininet testbed\\cite{lantz2010network}. For the throughput test, we use two PCs, one running mininet and the other running NOX controller. Iperf \\cite{tirumala2005iperf} is adopted as our test tool.\n\n \\textbf{Comparsion Metrics.} We use the following metrics to quantify the results\\cite{motiwala2008path, wangfast}.\n\\begin{itemize}\n\\item \\emph{Recovery Ratio}.\nRecovery ratio is introduced to evaluate the capacity of network recovery to reset the connection between pairs of disconnected nodes, which can be defined as follows,\n\\begin{definition}\n\\textbf{(Recovery Ratio)}\n$$\\mbox{Recovery Ratio}= {\\frac{\\mbox{number of recovered paths}}{\\mbox{number of recoverable paths}}}$$\n\\end{definition}\nAs a result of multiple failures, the underlying topology may be divided into disconnected components, or the source (and\/or destination) of a certain flow becomes failed. Hence, we call a disconnected routing path as ``recoverable'' if both the end nodes are alive and they are not physically separated.\n\\item \\emph{Path stretch}. The detail definition can be found in \\cite{W13fcr}. Generally, a path with longer stretch requires more network resources. We adopt the notation of \\emph{stretch} to measure the ratio of alternate path length over the expected shortest path length.\n\\item \\emph{Controller overhead}. As the aforementioned idea of backup topology generation, we try to reduce the load of the controller by locally restoring the failed connection. The effectiveness of this approach is measured through the metric below.\n\\begin{definition}\n\\textbf{(Controller Overhead)}\nController overhead is defined as the proportion of reconnection requests that need to be processed by the controller.\n\\end{definition}\n\\item \\emph{Maximal Load}. As defined in Section \\ref{sec:splicing-module}, it quantifies the maximal load among nodes after a regional failure.\n\\end{itemize}\n\nWe compare our SDN-based Fast and Resilient Routing against Disaster (SDN-FRRD) approach with the following approaches proposed in the literature,\n \\begin{itemize}\n\\item MRC\\cite{kvalbein2006fast}. The MRC is designed for local fast recovery. We evaluate it to see if it's sufficient for regional failures.\n\\item SDN-MRC. We apply the MRC to our novel framework, by directly using the MRC in the Backup Topologies Generation module.\n\\item Path Splicing\\cite{motiwala2008path}. The advanced multipath routing algorithm, path splicing, is to random splicing routes in the data plane. The setting of Path Splicing is: using the same number of $k$ backup topologies as MRC and SDN-MRC, the link weight perturbation function is: $weight(i,j)=(degree(i)+degree(j))\/degree_{max}$ where $degree_{max}$ is the maximal node degree and $weight(i,j)$ ranges from $0$ to $2$.\n\\end{itemize}\n\\subsection{Evaluation Results}\n\\begin{figure*}[!htb]\n  \\centering\n  \\subfigure[Germany backbone]{\n    \\label{50:result2}\n    \\includegraphics[width=2.0 in]{germany50-recovery.eps}}\n     \\subfigure[Rand50]{\n    \\label{50:result3}\n    \\includegraphics[width=2.0 in]{rand50-50-recovery.eps}}\n    \\subfigure[Rand100]{\n    \\label{50:result3}\n    \\includegraphics[width=2.0 in]{rand100-50-recovery.eps}}\n      \\caption{Recovery Ratio vs. Number of backup topologies $k$ when the failure radius=50}\n  \\label{fig:recovery_result_radius_50}\n\\end{figure*}\n\n\\begin{figure*}[!htb]\n  \\centering\n  \\subfigure[Germany backbone]{\n    \\label{100:result2}\n    \\includegraphics[width=2.0 in]{germany-100-recovery.eps}}\n     \\subfigure[Rand50]{\n    \\label{100:result3}\n    \\includegraphics[width=2.0 in]{rand50-100-recovery.eps}}\n    \\subfigure[Rand100]{\n    \\label{100:result3}\n    \\includegraphics[width=2.0 in]{rand100-100-recovery.eps}}\n      \\caption{Recovery Ratio vs. Number of backup topologies $k$ when the failure radius=100}\n  \\label{fig:recovery_result_radius_100}\n\\end{figure*}\n\n\\begin{figure*}[!htb]\n  \\centering\n  \\subfigure[Germany backbone]{\n    \\label{stretch:result2}\n    \\includegraphics[width=2.0 in]{germany-50-stretch.eps}}\n     \\subfigure[Rand50]{\n    \\label{stretch:result3}\n    \\includegraphics[width=2.0 in]{rand50-50-stretch.eps}}\n    \\subfigure[Rand100]{\n    \\label{stretch:result3}\n    \\includegraphics[width=2.0 in]{rand100-50-stretch.eps}}\n      \\caption{Path Stretch vs. Number of backup topologies $k$ when the failure radius=50}\n  \\label{fig:stretch_result_radius_50}\n\\end{figure*}\n\n\\begin{figure*}\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=1.8 in]{RRDis.eps}\n\\caption{Recovery Ratio vs. Failure Radius}\n\\label{fig:recovery-rand100-asc}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=1.8 in]{overhead6.eps}\n\\caption{Controller Overhead}\n\\label{fig:controller-overhead}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=1.8 in]{LB.eps}\n\\caption{ML on the Germany backbone}\n\\label{fig:ufd}\n\\end{minipage}\\hfill\n\\begin{minipage}[t]{0.23\\linewidth}\n\\centering\n\\includegraphics[width=1.8 in]{bandwidth-1.eps}\n\\caption{Receive Rate on the Iperf client}\n\\label{fig:bandwidth_1}\n\\end{minipage}\n\\end{figure*}\n\\subsubsection{Recovery Ratio}\nFig. \\ref{fig:recovery_result_radius_50} shows the recovery ratio in term of the number of backup topologies $k$ in the three topologies when the radius of the regional damage is $50$. From the graph, we can see that both SDN-FRRD and SDN-MRC, that apply the SDN framework can steadily achieve more than $90\\%$ recovery ratio. Comparing to the MRC and the Path Splicing curves, clearly shows the effectiveness of our SDN framework.\n\nFig. \\ref{fig:recovery_result_radius_100} shows the recovery ratio when the radius of the regional failure is $100$. The trend of the curves is similar to the ones inFig. \\ref{fig:recovery_result_radius_50}. When the failure radius is $100$, the failure breaks more links than when the failure radius is 50. Thus the recovery ratio of the MRC, SDN-MRC, Path Splicing gets decreased. For example, the recovery ratio decrease by about $5\\%$ in Fig. \\ref{100:result2} compared to Fig. \\ref{50:result2}, and decreases by about $10\\%$ in Fig. \\ref{100:result3} compared to Fig. \\ref{50:result3}. However, we observe no significant decrease of the curve SDN-FRRD. This is because in the Backup Topologies Generation module (see Algorithm \\ref{algo:b_t_g}), we adopt the vulnerable area of a path and consider the distribution of the failure radius to generate backup routes, such that the recovery ratio is not significantly influence by the size of the regional failure. This can also be validated in Fig. \\ref{fig:recovery-rand100-asc}, where the recovery ratio remains above $95\\%$ even when the failure radius is $150$ in all the three topologies.\n\nSince the recovery ratio of the SDN-FRRD is almost about $100\\%$ and is steady when the number of the backup topologies $k$ is from $6$ to $15$. Since small $k$ already has satisfying performance of the recovery ratio, small values of $k$ is sufficient. Because larger $k$ means the backup tables would consume more switch resources, network operators who have a strict limitation of switch resources can consider choosing the smallest $k$.\n\\subsubsection{Stretch}\nFig. \\ref{fig:stretch_result_radius_50} shows the stretch in term of $k=6, 7, 8, 9$ in the three topologies. As we can see, in all the three topologies, about $90\\%$ of the stretch is below 1.5. Normally, larger $k$ means more redundancies, which can lead to smaller stretches. The four values of $k$ achieve approximately equal recovery ratio, larger $k$ tends to have smaller stretch. This can be seen, for example, in Fig. \\ref{stretch:result3}, the curve of $k=9$ is on the left side of the curve of $k=6$, which means a smaller stretch. This leads to a trade off between cost and performance at the initialization of network, i.e., operators who want to get a lower stretch can choose larger $k$, at the cost of more switch\/router routing tables consumptions.\n\\subsubsection{Controller Overhead}\nFig. \\ref{fig:controller-overhead} shows that when $k=6$, the controller overhead in terms of the failure radius. In all the radiuses, the controller only needs to handle about $40\\%$ of failures, which means the data plane has already handle more than $60\\%$ of the failures. As the failure radius grows bigger, the controller overhead has the trend to get heavier too. This is because when more links are destroyed, it is more difficult for the data plane to recover from the failure. Even when the failure radius is 150, about $60\\%$ are handled locally.\n\\subsubsection{Maximal Load}\nFig. \\ref{fig:ufd} shows the maximal load of the splicing actions generation algorithm (Algorithm \\ref{algo:s_a_g}), compared to the shortest path splicing actions generation. The shortest path splicing actions generation choose the path with the minimal path length between a reconnection request $(s, t)$ when multiple rerouting paths between them are available\\cite{xie2014designing}. It however does not consider the load distribution among nodes. The results of the ML reduction are normalized based on the result of ILP1. From the graph, we can see that our algorithm can reduce the ML. As the failure radius becomes bigger, the ML also gets bigger, which indicates that without consider the load distribution, the load imbalance among node gets more severer.\n\\subsubsection{Recovery Time}\nFig. \\ref{fig:bandwidth_1} shows the receive rate on the Iperf client. A region failure occurred between the Iperf server and client at 0.3s. Packets can not be handled locally by backup tables, thus are sent to the controller. The receive rate on the client did have a sharp reduction at 0.3s, but it recovered very fast after about 10ms. The recovery time in real scenarios differs, which depends largely on the round trip time between a switch and a controller.\n\\section{Related Work}\nThere are limited number of recent papers focusing on leveraging SDN for large-scale regional failures. Nguyen \\textit{et al.} \\cite{nguyen2013software} studied latency between a switch and a controller and confirmed the applicability of SDN on disaster-resilient WANs. Works in \\cite{sharma2012openflow,staessens2011software} studied using SDN to meet carrier-grade requirements and pointed out that the reactive approach may not be able to achieve sub-50ms recovery. However, the above works did not consider the heavy controller overhead and the consistent packet processing problem\\cite{katta2013incremental,peresini2013cpp}. To reduce the recovery time, Sgambelluri \\textit{et al.} \\cite{sgambelluri2013openflow} proposed the proactive segment protection. Kamamura \\textit{et al.} \\cite{kamamura2013autonomous} gave a prototype to achieve IP fast rerouting using backup tables via autonomous OpenFlow controllers. The proposed proactive recovery can significantly reduce the recovery time. But, in face of region failure scenarios, the performance may be significantly decreased since the flexibility of SDN's global view is not used.\n\\section{Conclusion}\\label{sec:conclusion}\nIn this paper, we propose a SDN based architecture to enhance the reliability of network against disaster failures. We propose our algorithms for geographic-based backup topologies generation and splicing considering the laod distribution among nodes, and implement our approach by utilizing multiple tables pipeline processing and fast failover group tables of OpenFlow. Experiments show that, by well pre-designed backup topologies protection, our fast restoration approach can efficiently use the redundancy to achieve high reachability and low stretch with low controller overhead. The load distribution after a regional is more even, compared to the previous splicing algorithm in \\cite{xie2014designing}.\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}