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+{"text":"\\section{Introduction}\n\nIn \\cite{Penrose}, Penrose proposed an inequality as a natural consequence of Cosmic Censorship. The original set-up of Penrose consists of a shell of dust collapsing at the speed of light. The null hypersurface swept by the incoming shell separates the spacetime into two components with the flat Minkowski metric inside. Outside the null shell, the metric is no longer flat. The spacetime is vacuum except for a delta distribution of the energy-momentum tensor of matter density supporting along the null hypersurface. \n\nThe Penrose inequality in this case reduces to a geometric inequality on a marginally trapped surface in the null hypersurface. The location and geometry of the marginally trapped surface depends on the matter density which can be arbitrarily prescribed. This inequality should hold for a general spacelike $2$-surface in the Minkowski spacetime with minimal convexity assumptions to guarantee the regularity of the null hypersurface at infinity. It was observed by Gibbons \\cite{Gibbons1} that the inequality is exactly the classical Minkowski inequality when the $2$-surface lies in an Euclidean hyperplane. Tod \\cite{Tod1, Tod2} studied the case when the $2$-surface lies in the past null cone of a point and derived it from the Sobolev inequality. \n\nThe classical Minkowski inequality was generalized to a mean convex and star-shaped surface using the method of inverse mean curvature flow (cf. \\cite{Guan-Li}). Very recently, Huisken \\cite{Huisken} showed that the assumption that $\\Sigma$ is star-shaped can be replaced by the assumption that $\\Sigma$ is outward-minimizing. \n\nIn \\cite{Gibbons2}, Gibbons proposed a reduction scheme to approach the Penrose inequality for general surfaces in the Minkowski spacetime. The idea is to project the $2$-surface orthogonally onto an Euclidean hyperplane and to relate to the Minkowski inequality of the projected surface. However, Gibbons' calculation contains a mistake and the validity of this inequality for general surfaces remains open, see Section 7.1 of \\cite{Mars}. See also the detailed description in \\cite{Mars-Soria}, where the Penrose inequality in the Minkowski spacetime is proven\nfor a large class of surfaces. \n\nIn \\cite{Wang-Yau1, Wang-Yau2}, the authors made use of  Gibbons' projection procedure in their definition of quasi-local mass. It turns out the term that is missing from Gibbons's calculation corresponds a connection $1$-form of the normal bundle with respect to a certain normal frame on the $2$-surface. This term does not vanish in general and is essential in the new definition of quasi-local mass in \\cite{Wang-Yau1, Wang-Yau2}.\n\nIn \\cite{Brendle-Hung-Wang}, a Minkowski type inequality for surfaces in the Anti-deSitter-Schwarzschild manifold was proved using the inverse mean curvature flow and a new Heintze-Karcher type inequality in \\cite{Brendle}. When the mass parameter is zero, this inequality implies the Penrose inequality for a $2$-surface that lies on the hyperbola of the Minkowski spacetime, see Section 8 of \\cite{Wang}. In this article, we propose a conjecture generalizing the Penrose inequality for surfaces in the Minkowski spacetime. More specifically, the ambient space is the Schwarzschild spacetime. We  prove that the inequality holds in following four cases: (1) when the surface lies in a totally geodesic time slice; (2) when the surface lies in a totally umbilical slice; (3) when the surface lies in a null hypersurface emanating from a sphere of symmetry; (4) when the surface lies in a convex static timelike hypersurface (see Definition \\ref{convex.static} for a precise statement).\n\nWe remark that the Riemannian Penrose inequality was proved by Huisken-Ilmanen \\cite{Huisken-Ilmanen} and Bray \\cite{Bray}. For other related work on the Penrose inequality, we refer to \\cite{Mars} and references therein.\n\n\\section{Statement of the Penrose inequality}\n\n\\subsection{Minkowski spacetime}\nLet $\\Sigma$ be a two-dimensional spacelike closed  embedded orientable surface in the Minkowksi space $\\mathbb{R}^{3,1}$. Throughout the article, we assume $\\Sigma$ is diffeomorphic to $S^2$. We consider a fixed future timelike vector $T_0 \\in \\mathbb{R}^{3,1}$ satisfying $\\langle T_0,T_0 \\rangle = -1$. \n\nWe recall the mean curvature vector field $\\vec{H}$ of $\\Sigma$, which is the unique normal vector field such that the variation of area of $\\Sigma$ in a normal variation field $V$ is given by $-\\int_\\Sigma \\langle \\vec{H},V \\rangle \\, d\\mu$. The convention we adopt here makes the mean curvature vector of a standard round sphere inward pointing.  Let $L$ and $\\underline{L}$ be two null normals of $\\Sigma$ with $\\langle L, \\underline{L}\\rangle=2$. We assume $L$ is future-directed and $\\underline{L}$ is past-directed (both outward pointing whenever this makes sense). In terms of $L $ and $\\underline{L}$, we have \n\\[\\vec{H} = \\frac{1}{2} \\, (\\langle \\vec{H}, L\\rangle \\, \\underline{L} + \\langle \\vec{H}, \\underline{L}\\rangle \\, L).\\]\n\nThe dual mean curvature vector $\\vec{J}$ is defined as \n\\[\\vec{J} = \\frac{1}{2} \\, (\\langle \\vec{H}, L \\rangle \\, \\underline{L} - \\langle \\vec{H}, \\underline{L} \\rangle \\, L). \\] \n$\\vec{J}$ satisfies $\\langle \\vec{J}, \\vec{J} \\rangle = -\\langle \\vec{H},\\vec{H} \\rangle$ and $\\langle \\vec{J}, \\vec{H} \\rangle = 0$. In fact, $\\vec{J}$ is uniquely characterized by these properties, up to a sign. The choice here makes $\\vec{J}$ a future timelike vector in case $\\vec{H}$ is inward spacelike.\n\nThe following inequality for spacelike $2$-surfaces in the Minkowski spacetime was proposed by Penrose in \\cite{Penrose}: \n\n\\begin{conjecture} (Penrose)\nSuppose that $\\Sigma$ is past null convex in the sense that the past null hypersurface generated by $\\Sigma$ is smooth. Then \n\\begin{equation} \n\\label{penrose.inequality}\n- \\int_\\Sigma \\langle \\vec{J},T_0 \\rangle \\, d\\mu \\geq \\sqrt{16\\pi \\, |\\Sigma|}. \n\\end{equation}\n\\end{conjecture}\n\nBy the divergence theorem, we have \n\\[\\int_\\Sigma \\langle \\vec{H},T_0 \\rangle \\, d\\mu = 0.\\] \nThis implies \n\\begin{equation}\n-\\int_\\Sigma \\langle \\vec{H},L \\rangle \\, \\langle \\underline{L},T_0 \\rangle \\, d\\mu = \\int_\\Sigma \\langle \\vec{H},\\underline{L} \\rangle \\, \\langle L,T_0 \\rangle \\, d\\mu = -\\int_\\Sigma \\langle \\vec{J},T_0 \\rangle \\, d\\mu.\n\\end{equation}\nThus, the inequality \\eqref{penrose.inequality} can be rewritten as \n\\begin{equation} \n-\\int_\\Sigma \\langle \\vec{H},L \\rangle \\, \\langle \\underline{L},T_0 \\rangle \\, d\\mu \\geq \\sqrt{16\\pi \\, |\\Sigma|}. \n\\end{equation}\nThis formulation is independent of the choice of $L$ and $\\underline{L}$ except the normalization $\\langle L,\\underline{L} \\rangle = 2$. If we choose $\\underline{L}$ such that $\\langle \\underline{L},T_0 \\rangle=1$, then the inequality \\eqref{penrose.inequality} can be written in the form \n\\begin{equation} \n\\label{null.exp}\n\\int_\\Sigma \\theta \\, d\\mu \\geq \\sqrt{16\\pi \\, |\\Sigma|}, \n\\end{equation}\nwhere $\\theta = -\\langle \\vec{H},L \\rangle$ corresponds to the future (outward) null expansion. Note that the inequality \\eqref{null.exp} is equivalent to the inequality (51) in \\cite{Mars}.\n\n\\subsection{Schwarzschild spacetime}\n\nThe Schwarzschild spacetime metric is given by \n\\begin{equation}\n\\label{Sch_coor}\n-(1-\\frac{2m}{r}) \\, dt^2 + \\frac{1}{1-\\frac{2m}{r}} \\, dr^2 + r^2 \\, g_{S^2}, \n\\end{equation}\nwhere $g_{S^2} = d\\theta^2 + \\sin^2\\theta \\, d\\phi^2$ is the round metric on $S^2$.\n\nLet $\\Sigma$ be a closed embedded orientable spacelike $2$-surface in the Schwarzschild spacetime. Let $L$ and $\\underline{L}$ be two null normals of $\\Sigma$ with $\\langle L,\\underline{L} \\rangle = 2$. Again we assume $L$ is future-directed and $\\underline{L}$ is past-directed. \n\nSince $\\frac{\\partial}{\\partial t}$ is a Killing field, we have \n\\[\\int_\\Sigma \\langle \\vec{H},\\frac{\\partial}{\\partial t} \\rangle \\, d\\mu = 0.\\]\nAs above, this implies \n\\[-\\int_\\Sigma \\langle \\vec{H},L \\rangle \\, \\langle \\underline{L},\\frac{\\partial}{\\partial t} \\rangle \\, d\\mu = \\int_\\Sigma \\langle \\vec{H},\\underline{L} \\rangle \\, \\langle L,\\frac{\\partial}{\\partial t} \\rangle \\, d\\mu = -\\int_\\Sigma \\langle \\vec{J},\\frac{\\partial}{\\partial t} \\rangle \\, d\\mu,\\]\nwhere \n\\[\\vec{J} = \\frac{1}{2} \\, (\\langle \\vec{H}, L \\rangle \\, \\underline{L} - \\langle \\vec{H}, \\underline{L} \\rangle \\, L)\\] \ndenotes the dual mean curvature vector.\n\n\\begin{conjecture}\n\\label{conjecture} \nLet $\\Sigma$ be a spacelike $2$-surface in the Schwarzschild spacetime. Suppose that the past null hypersurface generated by $\\Sigma$ is smooth. Then \n\\begin{equation}\n\\label{Sch.penrose1} \n-\\int_\\Sigma \\langle \\vec{J}, \\frac{\\partial}{\\partial t} \\rangle \\, d\\mu + 16\\pi m \\geq \\sqrt{16\\pi|\\Sigma|}. \n\\end{equation} \nHere $m$ is the total mass of the Schwarzschild spacetime.\n\\end{conjecture}\n\nOf course, an equivalent formulation is \n\\begin{equation} \n\\label{Sch.penrose2} \n-\\int_\\Sigma \\langle \\vec{H},L \\rangle \\, \\langle \\underline{L},\\frac{\\partial}{\\partial t} \\rangle \\, d\\mu + 16\\pi m \\geq \\sqrt{16\\pi|\\Sigma|}. \n\\end{equation}\nThe Schwarzschild spacetime belongs to the larger class of static spacetimes. We recall that a spacetime $S$ is \\textit{static} if the metric is of the form $-\\Omega^2 \\, dt^2+g_M$ where $g_M$ is a Riemannian metric on a 3-manifold $M$ and $\\Omega$ is a smooth function defined on $M$. \n\n\\begin{definition}\n\\label{convex.static}\nLet $B$ be a complete timelike hypersurface in a static spacetime $S$. We say that $B$ is \\textit{convex static} if $B =\\{(t, x) :  t\\in \\mathbb{R}, x\\in \\hat{\\Sigma} \\}$ for some 2-surface $\\hat{\\Sigma} \\subset M$, and the second fundamental form  $\\hat{h}_{ab}$ and the induced metric $\\hat{g}_{ab}$ of $\\hat{\\Sigma}$ in $M$ satisfies $\\hat{h}_{ab} \\geq \\Omega^{-1} \\, \\hat{\\nu}(\\Omega) \\, \\hat{g}_{ab} > 0$. Here, $\\hat{\\nu}$ denotes the outward-pointing unit normal to $\\hat{\\Sigma}$.\n\\end{definition}\n\nThe condition $\\hat{h}_{ab} \\geq \\Omega^{-1} \\, \\hat{\\nu}(\\Omega) \\, \\hat{g}_{ab} > 0$ has a natural geometric interpretation: it implies that the second fundamental form $\\Pi$ of the timelike hypersurface $B$ is nonnegative when evaluated at a null vector, i.e. $\\Pi(X, X)\\geq 0$ if $X$ is null and tangent to $B$. \n\nIn this paper, we prove that the inequality holds for a large class of spacelike $2$-surfaces in the Schwarzschild spacetime.\n\n\\begin{theorem} \nLet $\\Sigma$ be a closed embedded orientable spacelike $2$-surface in the Schwarzschild spacetime. The Gibbons-Penrose inequality \\eqref{Sch.penrose1} holds in the following cases:\n\\begin{enumerate}\n\\item $\\Sigma$ lies in a totally geodesic spacelike hypersurface and $\\Sigma$ is mean convex and star-shaped.\n\\item $\\Sigma$ lies in a totally umbilical (spherically symmetric) spacelike hypersurface and $\\Sigma$ is mean convex and star-shaped.\n\\item $\\Sigma$ lies in a outward directed null hypersurface emanating from a sphere of symmetry.\n\\item $\\Sigma$ lies in a convex static timelike hypersurface. \n\\end{enumerate}\n\\end{theorem}\n\nWe remark that by taking $m=0$, these give rise to the Penrose inequality in the Minkowski spacetime in the corresponding cases (see also \\cite{Wang}).\n\n\\section{Proof of the inequality in four cases}\n\n\\subsection{Surfaces in a totally geodesic time slice}\nWe first check the case when $\\Sigma$ lies in a totally geodesic time-slice ($t=0$) and thus the induced metric is \n\\[\\frac{1}{1-\\frac{2m}{r}} \\, dr^2+r^2 \\, g_{S^2}.\\] \nThe future timelike unit normal is given by $e_0 = \\frac{1}{\\sqrt{1-\\frac{2m}{r}}} \\, \\frac{\\partial}{\\partial t}$. Let $L = e_0+\\nu$ and $\\underline{L} = -e_0+\\nu$ be the two null normals where $\\nu$ is the outward unit normal of $\\Sigma$ in the time-slice. We compute $\\vec{H} = -H\\nu$ and $\\vec{J} = He_0$. This gives \n\\[-\\int_\\Sigma \\langle \\vec{J},\\frac{\\partial}{\\partial t} \\rangle \\, d\\mu = -\\int_\\Sigma H \\, \\langle e_0,\\frac{\\partial}{\\partial t} \\rangle \\, d\\mu = \\int_\\Sigma H \\, \\sqrt{1-\\frac{2m}{r}} \\, d\\mu.\\] \nThus, the inequality \\eqref{Sch.penrose1} in this case is equivalent to \n\\begin{equation} \n\\label{eq_totally_geodesic} \n\\int_\\Sigma H \\sqrt{1-\\frac{2m}{r}} \\, d\\mu + 16\\pi m \\geq \\sqrt{16\\pi|\\Sigma|}. \n\\end{equation} \nNotice that the horizon area $|\\partial M|$ equals $16\\pi m^2$, and the static potential for the Schwarzschild space-time is $\\sqrt{1-\\frac{2m}{r}}$. Hence, the inequality \\eqref{eq_totally_geodesic} follows from results in \\cite{Brendle-Hung-Wang}. (Theorem 1 in \\cite{Brendle-Hung-Wang} works for arbitrary negative cosmological constant, and the result needed here follows by sending the cosmological constant to $0$.)\n\n\\subsection{Surfaces in a totally umbilical slice}\n\nWe claim that the inequality in Theorem 1 of \\cite{Brendle-Hung-Wang} for surfaces in the Anti-deSitter-Schwarzschild manifold corresponds to inequality \\eqref{Sch.penrose1} for surfaces in a spherically symmetric umbilical slice of the Schwarzschild space-time. Let us recall the definition of the three-dimensional Anti-deSitter-Schwarzschild manifold \\footnote{The definition here is slightly different from \\cite{Brendle} and \\cite{Brendle-Hung-Wang}, as we use $2m$ as the mass parameter instead of $m$}. We fix a real number $m > 0$, and let $s_0$ denote the unique positive solution of the equation \n\\begin{equation} \n\\label{eq_s_0} \n1 - 2m \\, s_0^{-1} + \\lambda^2 s_0^2 = 0. \n\\end{equation} \nWe then consider the manifold $M = S^2 \\times [s_0,\\infty)$ equipped with the Riemannian metric \n\\[\\bar{g} = \\frac{1}{1 - 2m \\, s^{-1} + \\lambda^2 s^2} \\, ds^2 + s^2 \\, g_{S^2},\\]\nwhere $g_{S^2}$ is the standard round metric on the unit sphere $S^2$. The sectional curvatures of $(M,\\bar{g})$ approach $-1$ near infinity, so $\\bar{g}$ is asymptotically hyperbolic. Moreover, the scalar curvature of $(M,\\bar{g})$ equals $-6$. The boundary $\\partial M = S^2 \\times \\{s_0\\}$ is referred to as the horizon.\n\nNow we return to the Schwarzschild spacetime. Consider a function $\\rho=\\rho(s)$ that satisfies \n\\[\\rho'(s) = \\frac{\\lambda s}{(1-\\frac{2m}{s})\\sqrt{1-\\frac{2m}{s}+\\lambda^2 s^2}}\\] \nfor some constant $\\lambda>0$. Take the embedding of $(2m, \\infty)\\times S^2$ into the Schwarzschild space-time by $\\Psi(s,\\theta,\\phi) = (\\rho(s),s,\\theta,\\phi)$ in Schwarzschild coordinates $(t,r,\\theta,\\phi)$ and denote the image by $\\hat{M}=\\{(t,r,\\theta,\\phi): t=\\rho(s), r=s\\}$. \n\nSubstituting $t=\\rho(s)$ and $r=s$ in \\eqref{Sch_coor}, it follows that the induced metric on $\\hat{M}$ is given by\n\\[\\frac{1}{1-\\frac{2m}{s}+\\lambda^2 s^2} ds^2 + s^2 \\, g_{S^2},\\] \nwhich is isometric to the one on an Anti-deSitter-Schwarzschild three-manifold $M$.\n\n\n\\begin{remark}\nThe function $\\rho$ appears to be only defined on $(2m, \\infty)$. However, we can extend $\\hat{M}$ in an extension of Schwarzschild space-time so that the domain of definition of $s$ extends to $(s_0,\\infty)$ where $s_0$ is the unique positive root of $1-\\frac{2m}{s}+\\lambda^2 s^2$. We refer to Section 6 of \\cite{Mars} where such an extension is carried out by using advanced Eddington-Finkelstein coordinates. In any case, we shall denote by $M$ the one that is extended to $(s_0,\\infty)$ which is referred as the Anti-deSitter-Schwarzschild manifold in \\cite{Brendle-Hung-Wang}. Note that $\\hat{M} \\subset M$ is an isometric embedding.\n\\end{remark}\n\n\\begin{proposition} \nThe hypersurface $\\hat{M}$ is umbilical, i.e the second fundamental form is proportional to the induced metric. \n\\end{proposition}\n\n\\begin{proof} \nLet $b(s)=1-\\frac{2m}{s}$ and $f(s)=\\sqrt{1-\\frac{2m}{s}+\\lambda ^2 s^2}$. We have the following relation:\n\\[b^{-1}-(\\rho')^2 b = f^{-2}.\\] \nAn orthonormal coframe adapted to the hypersurface $\\hat{M}$ is given by \n\\[\\theta^0=\\frac{1}{\\sqrt{b^{-1}-b(\\rho')^2}} \\, (dt-\\rho' \\, dr) = f(s) \\, (dt-\\rho' \\, dr),\\]\n\\[\\theta^1=\\frac{1}{\\sqrt{b^{-1}-b(\\rho')^2}} \\, (b\\rho' \\, dt-b^{-1} \\, dr) = f(s) \\, (b\\rho' \\, dt-b^{-1} \\, dr),\\]\n\\[\\theta^2=s \\, d\\theta,\\] and\n\\[\\theta^3=s \\sin\\theta \\, d\\phi,\\] \nwhere $\\theta^0$ is the unit conormal that is dual to the unit future timelike normal \n\\begin{equation} \n\\label{unit_normal} \ne_0=\\frac{f(s)}{b(s)} \\, \\frac{\\partial}{\\partial t} + \\lambda s \\, \\frac{\\partial}{\\partial r}. \n\\end{equation}\nThe second fundamental form can be computed using this coframe and we derive\n\\[p_{11}=\\frac{d}{ds}(\\frac{b\\rho'}{\\sqrt{b^{-1}-b(\\rho')^2}}),\\] \n\\[p_{22}=p_{33}=\\frac{1}{s} \\, \\frac{b\\rho'}{\\sqrt{b^{-1}-b(\\rho')^2}}.\\]\nWe check that \n\\[\\frac{b\\rho'}{\\sqrt{b^{-1}-b(\\rho')^2}}=\\lambda s.\\] \nThus, $p_{11}=p_{22}=p_{33}=\\lambda$ and $\\hat{M}$ is umbilical.  \n\\end{proof}\n\n\n\n\\begin{proposition}\n\\label{Sigma.lies.in.an.umbilic.hypersurface}\nFor a spacelike $2$-surface $\\Sigma$ in $\\hat{M}$ that is mean convex and star-shaped, the inequality \\eqref{Sch.penrose1} holds.\n\\end{proposition}\n\n\\begin{proof} \nWe assume $\\lambda=1$ for simplicity. (The general case can be reduced to this special case by scaling.) Consider a spacelike $2$-surface $\\Sigma$ in the umbilical hyersurface $\\hat{M}$. Let $\\nu$ be the outward unit normal of $\\Sigma$ in $\\hat{M}$, and let $L=e_0+\\nu$ and $\\underline{L}=-e_0+\\nu$ be the two null normals. The mean curvature vector $\\vec{H}$ is given by $-H\\nu + 2e_0$ where $H$ is the mean curvature of $\\Sigma$ in $\\hat{M}$ with respect to $\\nu$. Consequently, the dual mean curvature vector is \n\\[\\vec{J} = \\langle \\vec{H},e_0 \\rangle \\, \\nu - \\langle \\vec{H},\\nu \\rangle \\, e_0 = He_0 - 2\\nu.\\] \nThis implies \n\\[-\\int_\\Sigma \\langle \\vec{J},\\frac{\\partial}{\\partial t} \\rangle \\, d\\mu = -\\int_\\Sigma H \\, \\langle e_0,\\frac{\\partial}{\\partial t} \\rangle \\, d\\mu + 2 \\int_\\Sigma \\langle \\nu,\\frac{\\partial}{\\partial t} \\rangle \\, d\\mu.\\] \nAs above, we identify the hypersurface $\\hat{M}$ with a region in the three-dimensional Anti-deSitter-Schwarzschild manifold. The function \n\\[-\\langle e_0,\\frac{\\partial}{\\partial t} \\rangle = \\theta^0(\\frac{\\partial}{\\partial t}) = \\sqrt{1-\\frac{2m}{s}+\\lambda^2 s^2} = f(s)\\] \nis exactly the static potential for the Anti-deSitter-Schwarzschild space-time. Let us denote by $(\\frac{\\partial}{\\partial t})^\\top$ the component of $\\frac{\\partial}{\\partial t}$ that is tangential to $\\hat{M}$. From \\eqref{unit_normal} and $\\Psi_*(\\frac{\\partial }{\\partial s})=\\rho'(s)\\frac{\\partial}{\\partial t}+\\frac{\\partial}{\\partial r}$, we derive \n\\[(\\frac{\\partial}{\\partial t})^\\top = -s f(s) \\Psi_*(\\frac{\\partial}{\\partial s}),\\] \nhence \n\\[\\langle \\nu,\\frac{\\partial}{\\partial t} \\rangle = -\\langle \\nu,s \\, f(s) \\, \\Psi_*(\\frac{\\partial}{\\partial s}) \\rangle.\\] \nPutting these facts together, we obtain \n\\[-\\int_\\Sigma \\langle \\vec{J},\\frac{\\partial}{\\partial t} \\rangle \\, d\\mu = \\int_\\Sigma H \\, f \\, d\\mu - 2 \\int_\\Sigma \\langle \\nu,s f(s) \\frac{\\partial}{\\partial s} \\rangle \\, d\\mu.\\] The can be viewed as an equation on $M$ through the isometry. \nRecall $(M, \\bar{g})$ is defined for $s\\in [s_0, \\infty)$ with \n\\[\\bar{g}=\\frac{1}{f(s)^2} \\, ds^2+s^2 \\, g_{S^2}.\\]\nApplying the divergence theorem on $M$ gives\n\\[\\int_\\Sigma \\langle \\nu,s f(s) \\frac{\\partial}{\\partial s} \\rangle \\, d\\mu = \\int_\\Omega \\text{\\rm div}_{\\bar{g}} (s f(s)\\frac{\\partial}{\\partial s}) \\, d\\text{\\rm vol} + \\int_{\\partial M} \\langle \\nu,s f(s)\\frac{\\partial}{\\partial s}  \\rangle \\, d\\mu\\] \nwhere $\\partial M$ is the horizon and $\\Omega$ is the region enclosed by $\\partial M$ and $\\Sigma$.  A straightforward  computation shows  \n\\[\\text{\\rm div}_{\\bar{g}}(sf \\frac{\\partial}{\\partial s}) = 3f.\\] \nIn fact, $s f(s) \\frac{\\partial}{\\partial s}$ is the conformal Killing field used in \\cite{Brendle}.\nOn the other hand, on a level surface of $s$, $\\nu=f(s) \\, \\frac{\\partial}{\\partial s}$ and $\\langle \\nu,sf(s)\\frac{\\partial}{\\partial s} \\rangle = s$. Taking the limit $s \\searrow s_0$, we obtain \n\\[\\int_{\\partial M} \\langle \\nu,sf(s)\\frac{\\partial}{\\partial s} \\rangle \\, d\\mu = 4\\pi s_0^3.\\]\nIn summary, we have shown that \n\\[\\int_\\Sigma \\langle \\nu,s f(s) \\frac{\\partial}{\\partial s} \\rangle \\, d\\mu = \\int_\\Omega 3f \\, d\\text{\\rm vol} + 4\\pi s_0^3,\\] \nhence \n\\[-\\int_\\Sigma \\langle \\vec{J},\\frac{\\partial}{\\partial t} \\rangle \\, d\\mu = \\int_\\Sigma H \\, f \\, d\\mu - \\int_\\Omega 6f \\, d\\text{\\rm vol} - 8\\pi s_0^3.\\] \nNow  recall from \\cite{Brendle-Hung-Wang} that for such a surface in the Anti-deSitter-Schwarzschild space $M$,\n\\[\\int_\\Sigma f H \\, d\\mu - 6\\int_\\Omega f \\, d\\text{\\rm vol} \\geq \\sqrt{16\\pi|\\Sigma|}-8\\pi s_0.\\] \nTherefore, inequality \\eqref{Sch.penrose1} follows by combining the last two inequalities and the defining equation \\eqref{eq_s_0} of $s_0$,  which implies $s_0^3+s_0=2m$.\n\\end{proof}\n\n\\subsection{Surfaces in a null cone}\nLet $\\Sigma$ be a spacelike $2$-surface which is contained in the null hypersurface \n\\[N = \\{(t,r,\\theta,\\phi): t=s+2m \\log (\\frac{s}{2m}-1), r=s, s>2m\\}.\\] \nLet $L$ and $\\underline{L}$ be the null normal vectors to $\\Sigma$. Note that the future outward null normal $L$ is tangential to the null hypersurface $N$. Since $\\Sigma$ is spacelike, $\\Sigma$ can be written as a radial graph \n\\[\\Sigma = \\{(t,r,\\theta,\\phi):  t=r+2m \\log (\\frac{r}{2m}-1),  r=u(\\theta,\\phi), (\\theta, \\phi)\\in S^2\\}\\] \nfor some function $u: S^2 \\to (2m,\\infty)$.\n\nFor each $\\lambda>0$, we denote by $\\rho_\\lambda$ the unique solution of the ODE \n\\[\\rho'(s) = \\frac{\\lambda s}{(1-\\frac{2m}{s})\\sqrt{1-\\frac{2m}{s}+\\lambda^2 s^2}}\\] \nsuch that $\\rho_\\lambda(4m)=4m$. It is easy to see that the functions $\\rho_\\lambda(s)$ converge smoothly to the function $s+2m \\log(\\frac{s}{2m}-1)$ as $\\lambda\\rightarrow \\infty$ for $s$ in compact subintervals of $(2m,\\infty)$. Let \n\\[\\hat{M}_\\lambda = \\{ (t, r, \\theta, \\phi): t=\\rho_\\lambda(s), r=s, s> 2m \\}.\\] \nWe have seen above that $\\hat{M}_\\lambda$ is an umbilic hypersurface which is isometric to the $3$-dimensional Anti-deSitter-Schwarzschild manifold. Moreover, the surface \n\\[\\Sigma_\\lambda = \\{(t,r,\\theta,\\phi): t=\\rho_\\lambda({r}), r=u(\\theta,\\phi), (\\theta, \\phi)\\in S^2\\}\\] \ncan be viewed as a star-shaped surface within the Anti-deSitter-Schwarzschild manifold $\\hat{M}_\\lambda$. \n\nAs $\\lambda \\to \\infty$, the hypersurfaces $\\hat{M}_\\lambda$ converge smoothly to the null hypersurface $N$. Moreover, the surfaces $\\Sigma_\\lambda$ converge smoothly to the original spacelike $2$-surface $\\Sigma$. In particular, the mean curvature vector of $\\Sigma_\\lambda$ converges to the mean curvature vector of $\\Sigma$ as $\\lambda \\to \\infty$, and the dual mean curvature vector of $\\Sigma_\\lambda$ converges to the dual mean curvature vector of $\\Sigma$. \n\nFinally, the unit normal vector field to $\\Sigma_\\lambda$ within $\\hat{M}_\\lambda$ converges to the future outward normal vector $L$ after suitable rescaling. Since the null expansion of $\\Sigma$ along $L$ is strictly positive, we conclude that the mean curvature of $\\Sigma_\\lambda$ (viewed as a hypersurface in $\\hat{M}_\\lambda$) is strictly positive when $\\lambda$ is sufficiently large. Therefore, Proposition \\ref{Sigma.lies.in.an.umbilic.hypersurface} implies that the Gibbons-Penrose inequality \\eqref{Sch.penrose1} holds for $\\Sigma_\\lambda$ when $\\lambda$ is sufficiently large. Taking the limit as $\\lambda \\to \\infty$, we conclude that the Gibbons-Penrose inequality \\eqref{Sch.penrose1} also holds for the original surface $\\Sigma$.\n\n\\subsection{Surfaces in a convex static timelike hypersuface}\nLet us consider a spacelike $2$-surface $\\Sigma$ in the Schwarzschild spacetime. For abbreviation, we put $\\hat{\\Sigma} = \\pi(\\Sigma)$, where $\\pi: (t,r,\\theta,\\phi) \\mapsto (r,\\theta,\\phi)$ denotes the projection to the $t=0$ slice along the Killing vector field $\\frac{\\partial}{\\partial t}$. Let us choose parametrizations $F$ and $\\hat{F}$ for $\\Sigma$ and $\\hat{\\Sigma}$ so that $F(x) = (\\tau(x), \\hat{F}(x))$. Clearly,  \n\\[\\frac{\\partial F}{\\partial x_a} = \\frac{\\partial \\hat{F}}{\\partial x_a} + \\frac{\\partial \\tau}{\\partial x_a} \\, \\frac{\\partial}{\\partial t}\\] \nHence, the induced metric on $\\Sigma$ is related to the metric on $\\hat{\\Sigma}$ by \n\\[\\hat{g}_{ab} = g_{ab} + f^2 \\, \\partial_a \\tau \\, \\partial_b \\tau,\\] \nwhere, as usual, $f = \\sqrt{1-\\frac{2m}{r}}$. This gives \n\\[\\hat{g}^{ab} = g^{ab} - \\frac{f^2 \\, g^{ac} \\, g^{bd} \\, \\partial_c \\tau \\, \\partial_d \\tau}{1 + f^2 \\, |\\nabla \\tau|^2},\\] \nwhere $|\\nabla \\tau|^2 = g^{ab} \\, \\partial_a \\tau \\, \\partial_b \\tau$.\n\nWe next relate the second fundamental form of $\\Sigma$ to the second fundamental form of the projected surface $\\hat{\\Sigma}$. Consider the timelike hypersurface $B = \\{(t,x): t \\in \\mathbb{R}, \\, x \\in \\hat{\\Sigma}\\}$, and let $\\nu$ denote the outward-pointing unit normal vector to $B$. We may extend $\\nu$ to a vector field defined in an open neighborhood of $B$ such that $[\\nu,\\frac{\\partial}{\\partial t}] = 0$ and $\\langle \\nu,\\frac{\\partial}{\\partial t} \\rangle = 0$. \n\nNote that $\\nu$ is a normal vector field along both $\\Sigma$ and $\\hat{\\Sigma}$. Moreover, we have \n\\begin{align*} \n\\langle \\frac{\\partial F}{\\partial x_a},D_{\\frac{\\partial F}{\\partial x_b}} \\nu \\rangle \n&= \\langle \\frac{\\partial \\hat{F}}{\\partial x_a},D_{\\frac{\\partial \\hat{F}}{\\partial x_b}} \\nu \\rangle + \\frac{\\partial \\tau}{\\partial x_a} \\, \\frac{\\partial \\tau}{\\partial x_b} \\, \\langle \\frac{\\partial}{\\partial t},D_{\\frac{\\partial}{\\partial t}} \\nu \\rangle \\\\ \n&+ \\frac{\\partial \\tau}{\\partial x_a} \\, \\langle \\frac{\\partial}{\\partial t},D_{\\frac{\\partial \\hat{F}}{\\partial x_b}} \\nu \\rangle + \\frac{\\partial \\tau}{\\partial x_b} \\, \\langle \\frac{\\partial \\hat{F}}{\\partial x_a},D_{\\frac{\\partial}{\\partial t}} \\nu \\rangle \\\\ \n&= \\langle \\frac{\\partial \\hat{F}}{\\partial x_a},D_{\\frac{\\partial \\hat{F}}{\\partial x_b}} \\nu \\rangle + \\frac{\\partial \\tau}{\\partial x_a} \\, \\frac{\\partial \\tau}{\\partial x_b} \\, \\langle \\frac{\\partial}{\\partial t},D_\\nu \\frac{\\partial}{\\partial t} \\rangle \\\\ \n&+ \\frac{\\partial \\tau}{\\partial x_a} \\, \\langle \\frac{\\partial}{\\partial t},D_{\\frac{\\partial \\hat{F}}{\\partial x_b}} \\nu \\rangle + \\frac{\\partial \\tau}{\\partial x_b} \\, \\langle \\frac{\\partial \\hat{F}}{\\partial x_a},D_\\nu \\frac{\\partial}{\\partial t} \\rangle \\\\ \n&= \\hat{h}_{ab} - \\frac{\\partial \\tau}{\\partial x_a} \\, \\frac{\\partial \\tau}{\\partial x_b} \\, f \\, \\nu(f),\n\\end{align*} \nwhere $\\hat{h}_{ab}$ is the second fundamental form of the projected surface $\\hat{\\Sigma}$. This implies \n\\begin{align*} \n-\\langle \\vec{H},\\nu \\rangle \n&= g^{ab} \\, \\langle \\frac{\\partial F}{\\partial x_a},D_{\\frac{\\partial F}{\\partial x_b}} \\nu \\rangle \\\\ \n&= g^{ab} \\, \\hat{h}_{ab} - |\\nabla \\tau|^2 \\, f \\, {\\nu}(f) \\\\ \n&= \\hat{g}^{ab} \\, \\hat{h}_{ab} + \\frac{f^2 \\, g^{ac} \\, g^{bd} \\, \\partial_c \\tau \\, \\partial_d \\tau}{1 + f^2 \\, |\\nabla \\tau|^2} \\, \\hat{h}_{ab} - |\\nabla \\tau|^2 \\, f \\, \\nu(f) \\\\ \n&= \\hat{H} + \\frac{f^2 \\, g^{ac} \\, g^{bd} \\, \\partial_c \\tau \\, \\partial_d \\tau}{1 + f^2 \\, |\\nabla \\tau|^2} \\, (\\hat{h}_{ab} - f^{-1} \\, \\nu(f) \\, \\hat{g}_{ab}), \n\\end{align*} \nwhere $\\hat{H} = \\hat{g}^{ab} \\, \\hat{h}_{ab}$ denotes the mean curvature of $\\hat{\\Sigma}$. If $B$ is convex static in the sense of Definition \\ref{convex.static}, then the tensor $\\hat{h}_{ab} - f^{-1} \\, \\nu(f) \\, \\hat{g}_{ab}$ is positive semidefinite, and we obtain \n\\[-\\langle \\vec{H},\\nu \\rangle \\geq \\hat{H}.\\] \nOn the other hand, we have \n\\[-\\langle \\vec{J},\\frac{\\partial}{\\partial t} \\rangle = -\\langle \\vec{H},\\nu \\rangle \\, \\sqrt{-\\langle \\Big ( \\frac{\\partial}{\\partial t} \\Big )^\\perp,\\Big ( \\frac{\\partial}{\\partial t} \\Big )^\\perp \\rangle} = -\\langle \\vec{H},\\nu \\rangle \\, f \\, \\sqrt{1 + f^2 \\, |\\nabla \\tau|^2},\\] \nwhere $|\\nabla \\tau|^2 = g^{ab} \\, \\partial_a \\tau \\, \\partial_b \\tau$. Putting these facts together, we obtain the pointwise inequality \n\\[-\\langle \\vec{J},\\frac{\\partial}{\\partial t} \\rangle \\geq \\hat{H} \\, f \\, \\sqrt{1 + f^2 \\, |\\nabla \\tau|^2}.\\] \nThe volume elements of $\\Sigma$ and $\\hat{\\Sigma}$ are related by $d\\hat{\\mu} =\\sqrt{1+f^2 \\, |\\nabla\\tau|^2} \\, d\\mu$. Hence, integrating the last equation gives \n\\[-\\int_\\Sigma \\langle \\vec{J}, \\frac{\\partial}{\\partial t}\\rangle \\, d\\mu \\geq \\int_{\\hat{\\Sigma}} \\hat{H} \\, f \\, d\\hat{\\mu}.\\] \nOn the other hand, since $B$ is convex static, the surface $\\hat{\\Sigma}$ is star-shaped and convex. Using our results above, we obtain \n\\[\\int_{\\hat{\\Sigma}} \\hat{H} \\, f \\, d\\hat{\\mu}\\geq \\sqrt{16\\pi|\\hat{\\Sigma}|} - 16\\pi m.\\] \nHence, the desired inequality follows by observing that $|\\hat{\\Sigma}| \\geq |\\Sigma|$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\IEEEPARstart{S}{ingle} image super-resolution (SISR)~\\cite{zhang2015revisiting,zhang2016joint,liang2015incorporating} aims to recover a high-resolution (HR) image from the corresponding low-resolution (LR) image. It is a very practical technique due to its high value in various fields, such as producing high-definition images from low-cost image sensors, medical imaging and satellite imaging. Restoring the HR image from the single LR input is also a very difficult problem of high theoretical values which arouses more and more interests from the academic communities~\\cite{timofte2016seven,dong2016image,yang2014singleBenchmark,huang2015single} and large companies~\\cite{romano2017raisr,ledig2017photo,shi2016real}. Typically, it is very challenging to restore the missing pixels from an LR observation since the number of pixels to be estimated in the HR image is usually much larger than that in the given LR input. The ill-pose nature of single image super-resolution problem makes restoring HR images an arena to evaluate inference and regression techniques. Generally, SISR techniques can be roughly divided into three categories: the interpolation methods, the reconstruction methods~\\cite{irani1993motion} and the example based methods~\\cite{freeman2000learning,yang2008image}.\n\nMost of the recent SISR methods fall into the example based methods which learn prior knowledge from LR and HR pairs, thus alleviating the ill-posedness of SISR. Representative methods mainly rely on different learning techniques to incorporate image priors for super-resolution process, including neighbor embedding regression~\\cite{chang2004super,timofte2013anchored,timofte2014a+}, sparse coding~\\cite{yang2008image,yang2010imageTip}, tree based regressions~\\cite{schulter2015fast,Salvador2015} and deep convolutional neural network (CNN)~\\cite{dong2014learning,dong2016image,liang2016incorporating,kim2016accurate}.\n\n\n\\begin{figure*}[th]\n     \\centering\n     \\footnotesize\n     \\includegraphics[width=0.850\\textwidth]{.\/fig\/model.pdf}\n     \\caption{The architecture of our residual model.}\n  \\label{fig:arch}\n\\end{figure*}\n\nAmong the above techniques, deep learning techniques especially deep CNN have largely promoted the state-of-the-art performances in SISR area. Dong \\textit{et al}. \\cite{dong2014learning} firstly proposed a deep convolutional neural network termed SRCNN with three convolutional layers for image super-resolution which gave the best practise at that time.\nLater, Dong \\textit{et al}. \\cite{dong2016image} extended SRCNN with larger filter sizes and filter numbers while kept the depth of CNN fixed to further improve the performance. They found that deeper models were hard to train and \\cite{dong2016image} failed to boost the performance by increasing the depth. Such findings indicate that deeper models are not suitable for image super-resolution, which is counter-intuitive as deeper models have been proved more effective in many tasks~\\cite{simonyan2014very,he2015deep,he2016identity}. Instead of directly predicting the HR output,\nKim \\textit{et al}. \\cite{kim2016accurate} proposed a very deep CNN (VDSR) of depth up to 20 by a large skip connection to predict the residual image, \\textit{i}.\\textit{e}., the high frequency of the HR image. VDSR surpasses SRCNN with a large margin which mainly benefits from two aspects: deeper architecture and predicting high frequency of images only which is called residual learning by \\cite{kim2016accurate}. \n\n\n\nAs demonstrated in~\\cite{kim2016accurate}, the SR results have been improved as the VDSR network goes deeper to a certain depth (20). Although VDSR has achieved impressive results, the plain structure of VDSR which simply stacks layers hampers the convergence of deeper architectures due to the gradient exploding\/vanishing problem. It would bring little improvement as the network goes deeper. Fortunately, the residual network~\\cite{he2015deep,he2016identity} has successfully addressed this issue. As a result, different from VDSR, this paper has designed a novel very deep residual convolutional neural network shown in Fig.~\\ref{fig:arch}. As LR image and target HR image are highly correlated, predicting high frequency of the image only is a kind of residual learning which largely lowers the price for training.\nA totally deep residual CNN will be expected to fully take advantage of the correlations between LR and HR images. Moreover, skip connections or identity mapping shortcuts in deep residual CNN would alleviate gradient vanishing\/exploding problem when the network becomes increasingly deeper.\n\nFor neural network it is known that mean shifts toward zero or centering the activations speeds up learning~\\cite{le1991eigenvalues,orr2003neural} by bringing the normal gradient closer to the unit natural gradient~\\cite{amari1998natural}. LeCun \\textit{et al}.~\\cite{le1991eigenvalues} justified the benefits of centering activation functions and advised to center the distribution of supervision information. The mean value of distribution for high frequency in the images is around zeros, while the mean of raw HR image pixels biases above zero. Thus, it is easy to understand that predicting residual images (high frequency) instead of HR images has largely improve the convergence of the network~\\cite{kim2016accurate}. Batch Normalization (BN)~\\cite{ioffe2015batch}\nalso aimed to center activations by reducing the internal covariate shift with extra moving average computations.\nRaiko \\textit{et al}.~\\cite{raiko2012deep} proved that shortcut connections made the Fisher information matrix closer to a diagonal matrix and standard gradient closer to the natural gradient which contributes to centering the outputs and slopes of the hidden layers in multi-layer perceptron network. Thus, identity mapping shortcuts naturally help the residual network center the activations.\n\nThe Batch Normalization (BN) layers in the conventional residual branches~\\cite{he2015deep,he2016identity} are abandoned in our proposed architecture as skip connections and predicting high frequency (with a zero mean) have ensured centering the activations if the network is not too deep.\nOur designed residual architectures which we refer to as ``SRResNetNB'' have consumed less computational resources and achieved better performances empirically.\n\nWhile very deep CNN model would increase the model capacity, on the other hand, it would introduce a huge number of parameters which is sometimes unacceptable for applications. Thus, when hardware resources are limited, a lightweight architecture using less parameters is essential for real word applications. In this paper, the `shape' of deep CNN has been investigated to largely reduce the parameter numbers. The `shape' of deep CNN refers to depth, all the filter sizes and numbers of each layer which decide sizes and numbers of feature maps in each layer to form a global shape. \nWith a residual architecture and lightweight `shape' design, the proposed model can not only achieve state-of-the-art PSNR and SSIM results for single image super-resolution but also produce visually pleasant results.\n\nA preliminary version of this work was presented earlier~\\cite{yang2017single}. The present work adds to the conference oral version in significant ways: first, different deep architectures of residual branches are explored to further conclude a principle of designing a deep network for image super-resolution. Second, a considerable new\nanalysis from the perspective of centering activations and ensemble behaviors of residual networks has been represented and intuitive explanations are supplied to the result. In particular, a strategy of gradually varying the `shape' of the residual network has been clarified in constructing a lightweight structure, based on the assumption that the residual network has been seen as an ensemble of relatively shallow networks with a large capacity~\\cite{Veit2016Residual}. Third, more detailed experiments are represented to design the structures and retrench the parameters of the residual model.\n\n\n\n\\section{Related Works}\n\nIn the pioneer work by Freeman~\\textit{et al}.~\\cite{freeman2000learning}, the co-occurrence priors were proposed that similar LR local structures often relate to similar HR local information. From LR and corresponding HR images, LR and HR examples (patches or sub images) could be extracted to form training databases. The mappings from LR to HR examples call for accurate regression methods to be applied. In fact, the learning based regression methods especially deep learning based methods have dominated the example based methods.\n\n\n\n\nSince the work of SRCNN~\\cite{dong2014learning}, deep CNNs have refreshed the state-of-the-art performances in super-resolution area. Simply elaborating the filter sizes and filter numbers for SRCNN~\\cite{dong2016image} had further improved the performance. Wang~\\textit{et al}.~\\cite{wang2015deep} designed the CNN architecture to incorporate the sparse coding prior based on the learned iterative shrinkage and thresholding algorithm (LISTA). With sparsity prior modeling, the performance boosted even with a model of smaller size compared with SRCNN.\n\nKim~\\textit{et al}.~\\cite{kim2016accurate} made a breakthrough for image super-resolution by predicting residual images and using a much deeper CNN termed VDSR up to 20 layers, which largely accelerated the speed of training and outperformed SRCNN presented by Dong~\\textit{et al}.~\\cite{dong2016image}. To ensure the fast convergence of deep CNN and avoid gradient vanishing or exploding, a much larger learning rate for training was cooperated with adjustable gradient clipping in VDSR training. VDSR is inspired by the merits of VGG net which attempts to train a thin deep network. However, this kind of plain networks are not easy to be optimized when they go even deeper as demonstrated by He~\\textit{et al}.~\\cite{he2015deep,he2016identity}.\n\nThe difficulties of training deeper plain networks were carefully analyzed by He~\\textit{et al}.~\\cite{he2015deep,he2016identity}. The degradation problem~\\cite{he2015deep} has been observed that the testing accuracy even the training accuracy becomes saturated then degrades rapidly as plain networks go deeper. This degradation is caused by the difficulties of training other than overfitting. It has been demonstrated that learning a residual function is much more easier than learning the original prediction function with very deep CNN. Residual networks with a surprising depth were designed for image classification problems with skip connections or identity mapping shortcuts. Later, a detailed analysis on the mechanisms of identity mapping in deep residual networks and a new residual unit design have been represented in~\\cite{he2016identity}.\n\n\n\nResidual network has also been applied in conjunction with perceptual loss~\\cite{johnson2016perceptual} to generate style transferred image and produce visual more pleasing HR images in large magnification factors. SRResNet~\\cite{ledig2017photo}, another famous concurrent work with us has also designed a residual network with skip connections for image super-resolution which serves as the generator network in a generative adversarial framework, termed SRGAN. To produce photo-realistic results, SRGAN~\\cite{ledig2017photo} exploited an adversarial loss to replace the traditional mean squared reconstruction error. This adversarial framework recovered images with better perceptual quality and especially favored large upscaling factors (\\textit{e}.\\textit{g}., 4). The success of these work and our previous version~\\cite{yang2017single} has indicated the importance of skip-connections for image super-resolution. Later, Tai \\textit{et al}.~\\cite{tai2017image} introduced skip connections of multiple paths and shared the weights of residual units in one recursive block. Most of the residual networks for image super-resolution designed the residual branches as a combination of convolution, nonlinear activation (such as ReLU or PReLU) and Batch Normalization (BN) layers, which are the same as the residual branches for image classification task.\n\nThe idea of shortcuts has been related with centering the activations at zero for multi-layer perceptron network~\\cite{raiko2012deep}. Raiko \\textit{et al}.~\\cite{raiko2012deep} proposed to transform the outputs of each hidden layers in multi-layer perceptron network to have zero output and zero slope on average and use separate shortcut connections to model the linear dependencies. It is known that centering the activations accelerates learning~\\cite{clevert2015fast,raiko2012deep,le1991eigenvalues,orr2003neural}. LeCun \\textit{et al}.~\\cite{le1991eigenvalues} analyzed the eigenvalues of Hessian matrix during the gradient descend process and give a theoretical justification for the benefits of centering activation functions.\n\nThe applied skip-connections have already centered the activations at zero within certain depth and the mean of distributions for high frequencies in images is close to zero, which indicate the BN layers could be eliminated in our residual units.\n\nOur design has been further supported by the very recent work~\\cite{lim2017enhanced}, which wons the first prize in Ntire 2017 challenge on single image super-resolution~\\cite{timofte2017ntire}. Liang \\textit{et al}.~\\cite{liang2017single} further extended the identity skip connections to projection skip connections and explored the power of internal priors for deep architectures.\n\n\nAfter largely easing the difficulties of training much deeper CNN with residual functions by shortcuts or skip connections, the huge number of parameters in deep architecture is still a big problem for computational resources and storages. The evolvement of Inception models~\\cite{szegedy2015going,ioffe2015batch,szegedy2016rethinking,szegedy2017inception} has demonstrated that carefully designed topologies enable compelling performances with less parameters. He \\textit{et al}.~\\cite{he2015deep,he2016identity} attempt to alleviate the problem by bottleneck architectures.\nThe bottleneck architectures first utilize $1\\times1$ convolutions to reduce the dimensions, then after some operations, $1\\times1$ convolutions are applied again to increase the dimensions. With such a residual unit design, the number of parameters could be largely reduced. Thus, the `shape' of CNN could be potentially explored to reduce the parameters while maintain the performances. The bottleneck architectures decrease the parameter numbers at the expense of increasing the depth of the network to mountain the performances. In the meanwhile, contextual information is very important for image super-resolution~\\cite{kim2016accurate,dong2016image}, such $1\\times1$ convolutions design may give a negative effort to the SR results.\nA study on the influences of the `shape' (the filter sizes, depth and numbers of convolutions in each layer) on the performance of image super-resolution has been represented in the following sessions.\nWith a carefully design and exploration of the `shape' of the network, novel residual deep models are proposed for image super-resolution task in this paper.\n\n\n\\section{A lightweight Residual Deep Model for Image Super-resolution}\n\nFollowing the example based methods, HR examples $I^h$ and LR examples $I^l$ are extracted from HR images $I^H$ and LR images $I^L$ respectively.\nThe degeneration process of LR images $I^L$ from the corresponding HR images $I^H$ could be considered as the following blurring process related with blur kernel $G$ and downsampling process $\\downarrow_{s}$ with a a scale factor $s$\n            \\begin{equation}\\label{eq:downsample}\n                 I^L=(I^H\\otimes{G})\\downarrow_{s}.\n             \\end{equation}\nIn the experiments, this process is simulated by a `bicubic' downscale interpolation.\n\n\n\nIn the next part, residual deep models for image super-resolution will be designed from the perspective of centering the activations to speed up learning.\n\n\\begin{figure*}[!htb]\n     \\centering\n     \\footnotesize\n     \\includegraphics[width=0.750\\textwidth]{.\/fig\/Visio-res_unit.pdf}\n     \\caption{The architectures of different residual units.}\n  \\label{fig:res-unit}\n  \\end{figure*}\n\n\n\n\n\n\\subsection{Deep Residual Models}\n\nThe gradient exploding\/vanishing problems are largely alleviated by skip connections in the deep residual models~\\cite{he2015deep,he2016identity}. The architectures of our deep residual models especially the residual branches will be further designed from the perspective of centering activations.\n\nSimply stacking the convolutional layers and rectified linear units as VDSR fashions~\\cite{kim2016accurate} will have a mean activation larger than zero~\\cite{clevert2015fast}. The non-zero mean activation acts as bias for the next layer. The more the layers are correlated, the higher their bias shift.\n\n\n\nFor a multi-layer perceptron network, Raiko \\textit{et al}.~\\cite{raiko2012deep} proved that the transformation by shortcuts centered the activations which made the Fisher information matrix closer to a diagonal matrix, and thus standard gradient closer to the natural gradient. The transformations can be as\n\\begin{equation}\\label{eq:transform}\n                  x^{k+1}=\\mathbf{A}\\cdot \\mathrm{T}(\\mathbf{B}\\cdot x^k)+\\mathbf{C}\\cdot x^k,\n             \\end{equation}\nwhere $\\mathbf{A}$, $\\mathbf{B}$, $\\mathbf{C}$ is the weight matrices, $\\mathrm{T}$ is a nonlinearity activation, $x^k$ is the output of the neurons of the kth layer.\n\nSimilarly, for convolutional neural network, transformations can be as\n\\begin{equation}\\label{eq:trans-conv}\n                  x^{k+1}=f(\\theta^k,x^k)+\\mathbf{C}\\cdot x^k,\n             \\end{equation}\nwhere $f$ is a function composed by convolutions, nonlinearity activation, and Batch Normalization (BN). When the weight matrix $\\mathbf{C}$ becomes identity matrix, function $f(\\theta^k,x^k)$ will become our residual branches. Thus, our residual networks with skip connections can naturally centering activations and speed up learning.\n\nFor image super-resolution problems, super-resolution is only applied on the luminance channel (Y channel in YCbCr color space) in most of previous study~\\cite{timofte2013anchored,dong2014learning,kim2016accurate}. It is obvious that the distribution of values on the luminance channel in the output HR images doesn't center at zero, while the residual images (high frequency of the images) have means towards zero. To center the activations, our deep residual CNN applies a large skip-connection as \\cite{kim2016accurate} which makes the network predict the residual images (the high frequency of the images). Predicting the residual images has largely improved the training speed and convergency results.\n\nOur deep residual CNN for image super-resolution is an end-to-end mapping model which can be roughly divided into three sub-networks to perform three steps: feature representation, nonlinear mapping, and reconstruction.\n\nThe feature representation sub-network extracts LR discriminative features from the LR input images, while nonlinear mapping part maps the LR feature representations into HR feature representations. Reconstruction part restores the HR images from HR feature representations. Feature representation sub-network applies plain network stacking convolutional and ReLU layers as shown in Fig.~\\ref{fig:arch} and reconstruction sub-network only uses convolutional layers as~\\cite{liang2017single}. The main body of our model, nonlinear mapping part consists of residual units which center the activations with shortcuts and ease the difficulties of training.\n\nTypical units of our deep residual CNN are shown in Fig.~\\ref{fig:res-unit}. Empirically, residual unit with 2 or 3 convolutional layers works well for image super-resolution problem, those two kinds of units are applied in the experiments. When featuremap dimensions change, the identity shortcut becomes a projection to change feature dimensions. The second right and rightmost are one unit of residual net for image classification problems proposed by He~\\textit{et al}.~\\cite{he2015deep,he2016identity} respectively. Compared with them, the architectures of our residual functions are composed of convolutional, ReLU layers and shortcuts, which are quite different. Batch Normalization units are discarded and deployments are different. Batch Normalization~\\cite{ioffe2015batch} reduces the distribution variations of layers (internal covariate) by normalizing the input of the layers. With an input x, the output of BN layer is given by\n           \\begin{equation}\\label{eq:res-f}\n                     BN_{\\gamma,\\beta}=\\gamma (\\frac{x-\\mu}{\\sqrt{\\sigma^2+\\varepsilon}})+\\beta,\n             \\end{equation}\nwhere $\\gamma$ and $\\beta$ are learnable parameters,  $\\mu$ and $\\sigma$ are the mean and variance of activations in the mini-batch, respectively, $\\varepsilon$ is a small constant for numerical stability. Obviously, the activation after Batch Normalization operation has also been centered.  As skip connections (Eq.~\\eqref{eq:trans-conv}) have naturally corrected the bias shift, thus if the residual network is not that deep\\footnote{The bias from zero will accumulate as the network goes deeper.}, the BN layers can be abandoned as it needs extra learning and inference computations which take much more computational resources.\n\nShortcuts or skip connections which are identity mappings are realized by element-wise additions. As this element-wise addition increases very little computations, our feed-forward deep residual CNN has a similar computational complexity with VDSR~\\cite{kim2016accurate} fashions network. Similar with {VDSR}~\\cite{kim2016accurate}, small convolutional filter of size $3\\times3$ has been applied. Assuming the input of $k$-th residual unit as $x^k$, the residual functions have the following form\n             \\begin{equation}\\label{eq:res-f}\n                     x^{k+1}=x^k+f(\\theta^k,x^k),\n             \\end{equation}\nwhere $\\theta^k$ are the parameters of $k$-th residual unit.\n\n \nA simple Euclidean loss function is adopted to make predictions approximate the high frequencies of examples\n  \\begin{equation}\\label{eq:hf}\n                 \\mathcal{L}=\\frac{1}{2n}\\sum_{i=1}^n\\|\\mathcal{F}(\\theta,I_i^l)-(I_i^h-I_i^l)\\|^2\n             \\end{equation}\nwhere $n$ is the number of patch pairs $(I^l, I^h)$, $F(\\theta,I^l)$ denotes the predictions of our deep residual CNN with parameter $\\theta$. Our deep residual CNN is composed of several \\textbf{Container}s which have certain number of residual units. For succinctness, the filter numbers keep the same in each single container. The architectures of our deep residual CNN will be described as a sequence of the filter numbers (${N1}_{k1}$, ${N2}_{k2}$, $\\cdots$) in containers. If subscript $k$ exists for $N_k$, it means there are $k$ residual units with each having a filter number of $N$ in this container.\n\n   Stochastic gradient descent (SGD) with the standard back-propagation \\cite{krizhevsky2012imagenet} is applied to train our deep residual CNN. In particular, the parameter is updated as Eq.~\\eqref{eq:sto_Grad}, where $m$ denotes the momentum parameter with a value of 0.9 and $\\eta$ is the learning rate.\n                \\begin{equation}\n                \\label{eq:sto_Grad}\n                   \\triangle_{i+1}= m\\cdot\\triangle_{i}+ \\eta\\cdot\\frac{\\partial loss}{\\partial \\theta_{i}},\\quad \\theta_{i+1}= \\theta_{i}+\\triangle_{i+1}\n                \\end{equation}\n\n High learning rates are expected to boost training with faster and better convergency. Adjustable gradient clipping \\cite{kim2016accurate} is utilized to keep learning rates high while at the same time to prevent the net from gradient exploding problems. Gradients $\\frac{\\partial Loss}{\\partial \\theta_{i}}$ are clipped into the range of $[-\\frac{\\tau}{\\eta},\\frac{\\tau}{\\eta}]$, where $\\tau$ is a constant value.\n\n\n\n\n\\subsection{Lightweight Design for the Proposed Model}\n\nIn this section, the `shape' of deep CNN has been explored to achieve better performances but with less number of parameters. The `shape' of deep CNN is determined by all the sizes and numbers of filters in each layer besides the depth of the network. Thin but small filter size works well with padding which leads to larger receptive field as network goes deep, in specific, $3\\times3$ filter size has been applied. It is general that deeper and wider network will have larger model capacity and better feature representational ability. However, the number of parameters is restricted by the hardware or computational resources. Using less parameters to achieve better performances is essential for applications. Next, filter numbers and the combinations of filter numbers will be discussed to retrench parameters for a better performance.\n\n\\begin{figure*}[bhtp]\n     \\centering\n     \\footnotesize\n     \\includegraphics[width=0.800\\textwidth]{.\/fig\/residual-ensemble.pdf}\n     \\caption{Residual network behaves like an ensemble of networks}\n  \\label{fig:ensemble}\n\\end{figure*}\n\n\\subsubsection{Exploring the Shape of the Architecture}\n\nInspired by the evolvement of Inception models~\\cite{szegedy2015going,ioffe2015batch,szegedy2016rethinking,szegedy2017inception} and the bottle-neck architecture \\cite{he2016identity}, it is supposed that changing the shape of the architecture may maintain the performance while largely reduce the computational parameters. Instead of applying $1\\times1$ convolutions as bottle-neck architecture, the $3\\times3$ convolutions are applied as image SR process largely depends on the contextual information in local neighbor areas.\n\nThe filter numbers of VDSR are kept the same. There seems to be few principles to decide filter numbers and the combinations of filter numbers in a network. Instead of using a same number of filters in a network, the filter numbers can be varied to potentially reduce parameters which could enable a deeper or wider network.\n\nResidual networks can be interpreted as an ensemble of many paths of differing depth~\\cite{Veit2016Residual} and residual networks enable very deep networks by leveraging only the short paths during training~\\cite{Veit2016Residual}. According to this assumption, if the models of short paths in the residual network have been less disturbed, the performance of residual network which is an ensemble could keep stable.\n\nA strategy of gradually varying the `shapes' of residual models is proposed by us to reduce parameters. Gradually varying the shape of network means the filter numbers of the adjacent layers should increase or decrease gradually. This has been illustrated as Fig.~\\ref{fig:ensemble}. In Fig.~\\ref{fig:ensemble}, different residual branches and corresponding skip connections are denoted by different colors. The residual networks can be unfolded as a summations of models from different paths of residual networks. Considering a residual network with three units or sub residual network, if the filter numbers of the adjacent layers change gradually, \\textit{e}.\\textit{g}., only the filter numbers of R3 changes (\\textit{e}.\\textit{g}., decreases), a lot of paths are unaffected. Thus, the residual networks are more robust to the shape varying and our strategy can be applied to achieve better performances with less parameters.\n\n\\begin{figure*}[htb]\n    \\centering\n     \\setlength{\\tabcolsep}{2pt}\n  \\footnotesize\n  \\begin{tabular}{ccccc}\n\n  \\includegraphics[width=0.200\\textwidth]{.\/fig\/s_inc.pdf} &\n   \\includegraphics[width=0.200\\textwidth]{.\/fig\/s_dec.pdf} &\n   \\includegraphics[width=0.200\\textwidth]{.\/fig\/s_inc_dec.pdf} &\n   \\includegraphics[width=0.200\\textwidth]{.\/fig\/s_dec_inc.pdf} &\n   \\includegraphics[width=0.200\\textwidth]{.\/fig\/s_same.pdf} \\\\\n   (a)&(b)&(c)&(d)&(e)\\\\\n   \n  \\end{tabular}\n  \\caption{Different `shapes' of networks which gradually vary the feature map numbers. The width of the block correlates to the number of feature maps in the layer.}\n  \\label{fig:shapes}\n\\end{figure*}\n\n\n\nThe impacts of the feature map numbers in each layer on performance are carefully explored in the following fashions as Fig.~\\ref{fig:shapes}: gradually, the numbers of feature maps\n \\begin{itemize}\n   \\item increase monotonically up to N.\n   \\item decrease monotonically from N.\n   \\item  increase up to N then decrease.\n   \\item decrease from N then increase.\n   \\item keep the same as N (baseline).\n \\end{itemize}\nIn Fig.~\\ref{fig:shapes}, the width of the square block correlates to the numbers of the feature map in the layer. The larger width of square block indicates there are more feature maps in that layer. Compared with the baseline way that the feature map numbers keep the same, applying gradually varying the shape strategy has largely reduced the parameters.\n\n\nThe experiments demonstrate that different lightweight designs have achieved comparable performances with less parameters. This will be further discussed in the experiments part. In comparison with our residual CNN, the performances of {VDSR} with different shapes fluctuate heavily. This proves our residual architectures are more robust to the shape varying of CNN and our strategy of gradually varying the `shape' of residual network could be applied to achieve better performances with less parameters.\n\n \\subsubsection{Training with Multiple Upscaling Factors to Retrench Parameters}\n It has been pointed out that it is feasible to train a deep CNN for different upscaling factors~\\cite{kim2016accurate}. Training datasets for different specified upscaling factors are combined together to enable our deep residual CNN to handle multiple upscaling factors, as images across different scales share some common structures and textures. Parameters are shared across different predefined upscaling factors which further dispenses with the trouble of retaining different models for different upscaling factors. It will retrench parameters when multiple upscaling factors are required.\n\n\n  \n   \\begin{table*}[thb]\n        \\centering\n       \n        \n        \n         \\caption{Comparison in different datasets and with different scales.}\n         \\label{table:results}\n         \\begin{tabular}{c|c|c|c|c|c|c|c|c|c}\n         \\hline\n         \\multirow{2}{*}{Dataset} & \\multirow{2}{*}{Scale} & Bicubic & A+\\cite{timofte2014a+} & RFL\\cite{schulter2015fast} & SelfEx\\cite{huang2015single} & SRCNN\\cite{dong2014learning} & VDSR\\cite{kim2016accurate} & SRResNetNB&R-basic\\\\\n         && PSNR\/SSIM & PSNR\/SSIM & PSNR\/SSIM & PSNR\/SSIM & PSNR\/SSIM & PSNR\/SSIM & PSNR\/SSIM& PSNR\/SSIM\\\\\n         \\hline\n         \\multirow{3}{*}{Set5} & $\\times$2 & 33.66\/0.9299 & 36.54\/0.9544 & 36.54\/0.9537 & 36.49\/0.9537 & 36.66\/0.9542 & \\textbf{37.53}\/\\textbf{0.9587} &37.51\/\\textbf{0.9587}&  37.27\/0.9577\\\\\n         & $\\times$3 & 30.39\/0.8682 & 32.58\/0.9088 & 32.43\/0.9057 & 32.58\/0.9093 & 32.75\/0.9090 & 33.66\/0.9213 &\\textbf{33.72}\/\\textbf{0.9215}& 33.43\/0.9190\\\\\n         & $\\times$4 & 28.42\/0.8104 & 30.28\/0.8603 & 30.14\/0.8548 & 30.31\/0.8619 & 30.48\/0.8628 & 31.35\/\\textbf{0.8838} & \\textbf{31.37}\/\\textbf{0.8838}& 31.15\/0.8796\\\\\n         \\hline\n         \\multirow{3}{*}{Set14} & $\\times$2 & 30.24\/0.8688 & 32.28\/0.9056 & 32.26\/0.9040 & 32.22\/0.9034 & 32.42\/0.9063 & 33.03\/0.9124 & \\textbf{33.10}\/\\textbf{0.9131}&32.86\/0.9113  \\\\\n         & $\\times$3 & 27.55\/0.7742 & 29.13\/0.8188 & 29.05\/0.8164 & 29.16\/0.8196 & 29.28\/0.8209 & 29.77\/0.8314 & \\textbf{29.80}\/\\textbf{0.8317} & 29.67\/0.8297 \\\\\n         & $\\times$4 & 26.00\/0.7027 & 27.32\/0.7491 & 27.24\/0.7451 & 27.40\/0.7518 & 27.49\/0.7503 & 28.01\/0.7674 & \\textbf{28.06}\/\\textbf{0.7681} &27.90\/0.7648\\\\\n         \\hline\n         \\multirow{3}{*}{BSD100} & $\\times$2 & 29.56\/0.8431 & 31.21\/0.8863 & 31.16\/0.8840 & 31.18\/0.8855 & 31.36\/0.8879 & 31.90\/0.8960 & \\textbf{31.91}\/\\textbf{0.8961}&31.76\/0.8940 \\\\\n         &$\\times$3 & 27.21\/0.7385 & 28.29\/0.7835 & 28.22\/0.7806 & 28.29\/0.7840 & 28.41\/0.7863 & 28.82\/0.7976 & \\textbf{28.83}\/\\textbf{0.7980} &  28.73\/0.7954\\\\\n         &$\\times$4 & 25.96\/0.6675 & 26.82\/0.7087 & 26.75\/0.7054 & 26.84\/0.7106 & 26.90\/0.7101 & \\textbf{27.29}\/\\textbf{0.7251} & 27.27\/0.7248 &27.19\/0.7221 \\\\\n         \\hline\n         \\multirow{3}{*}{Urban100} & $\\times$2 & 26.88\/0.8403 & 29.20\/0.8938 & 29.11\/0.8904 & 29.54\/0.8967 & 29.50\/0.8946 & 30.76\/0.9140 & \\textbf{30.88}\/\\textbf{0.9150}&30.47\/0.9100   \\\\\n         & $\\times$3 & 24.46\/0.7349 & 26.03\/0.7973 & 25.86\/0.7900 & 26.44\/0.8088 & 26.24\/0.7989 & 27.14\/0.8279 & \\textbf{27.17}\/\\textbf{0.8283}& 26.92\/0.8208\\\\\n         & $\\times$4 & 23.14\/0.6577 & 24.32\/0.7183 & 24.19\/0.7096 & 24.79\/0.7374 & 24.52\/0.7221 & 25.18\/0.7524 & \\textbf{25.22}\/\\textbf{0.7537}&  25.02\/0.7452\\\\\n         \\hline\n         \\end{tabular}\n   \\end{table*}\n\n\\section{Experiments}\n\nIn this section, we conducted a series of experiments to explore the empirical principles to design a deep architecture for image super-resolution problem. The performances of the proposed method against the state-of-the-art SISR methods are compared which clearly demonstrate better or comparable subjective scores and more visual pleasing results.\n\nThe same 291 training images applied by VDSR were utilized for training, including 91 images proposed in Yang \\textit{et al}. \\cite{yang2008image} and 200 natural images from Berkeley Segmentation Dataset (BSD). For testing, four datasets were investigated: `Set5' and `Set14' \\cite{timofte2013anchored,dong2014learning},`Urban100' \\cite{huang2015single} and `BSD100' \\cite{timofte2013anchored,yang2014singleBenchmark}.\n\nThe size of example was set as $41\\times41$ and the batch size was chosen as 64. Momentum and weight decay parameters were fixed as $0.9$ and $0.0001$ respectively. Multi-scale training was applied in all of the following experiments. Weight initialization methods \\cite{he2015deep,he2016identity} were applied with small modulations. Learning rate was initially set to 0.1 and then decreased by a factor of 10 every 30 epochs. All these settings ensure us to make a fair comparison with the competing approaches including VDSR method.\n\n\\subsection{Comparisons with the State-of-the-art Methods}\n\nTable \\ref{table:results} shows the quantitative comparisons with A+~\\cite{timofte2014a+}, RFL~\\cite{schulter2015fast}, SelfEx~\\cite{huang2015single}, SRCNN~\\cite{dong2014learning} and VDSR~\\cite{kim2016accurate}. Visual results are also represented to give intuitive assessment. In Table~\\ref{table:results}, two models of our deep residual CNN with different depth have been investigated, denoted as \\textbf{R-basic} and \\textbf{SRResNetNB} respectively.  The residual unit in R-basic and deeper and larger model SRResNetNB has two convolutional layers. R-basic $(16_3,32_3,64_3)$ has 22 layers, while SRResNetNB $(16_3,32_3,64_3,128_3,256_3)$ has 34 layers. SRResNetNB has achieved the best performances compared with other methods in most cases and comparable results in other situations.\n\n   \n\nR-basic outperforms the other methods except VDSR. However, the performances of VDSR (20 layers)  have not been obtained by our reimplementation. For example, the average PSNR of VDSR by our reimplementation for Set5 and Set14 are 37.32dB and 32.89dB respectively, with a gap of more than 0.1db from the reported results. Assisted with the missing tricks, the performance of our model is expected to be further boosted. In Fig.~\\ref{fig:Comp_a}, the PSNR against training epochs has been compared among R-basic, SRResNetNB, and VDSR trained by us. Deeper and larger model SRResNetNB outperformed {VDSR} at very beginning with a large margin. Although R-basic contains much less parameters, R-basic model has obtained comparable performances with VDSR.\n\n\\begin{figure}[!thb]\n  \\centering\n  \\footnotesize\n    \\includegraphics[width=0.40\\textwidth]{.\/fig\/comp.pdf}\n  \\caption{Comparisons of test psnr on Set 14 against training epochs among SRResNetNB(denoted as R-deep), R-basic and VDSR.}\n  \\label{fig:Comp_a}\n\\end{figure}\n\nIn Fig.~\\ref{fig:SRresults}, all the compared results are obtained by the released code of the authors or from the reported ones in the paper. Visually pleasing results have been achieved by our model. Restorations of our method contain more authentic texture and more clear details compared with the results by other methods such as the texture of the zebra head. Our method has provided less artifacts, \\textit{e}.\\textit{g}., all the other methods except ours have restored obvious artifacts at the location of book. Shaper edges have appeared in our restorations which have represented visually more pleasing results.\n\n\n\\subsection{Number of Parameters}\n\nFor R-basic model, there are 22 convolutional layers and 0.3M(322721) parameters accumulated by the numbers of corresponding weights and bias. For SRResNetNB model, 34 convolutional layers and 5M(4975905) parameters are applied. The compared VDSR in Table \\ref{table:results} is 20 layers and has 0.7M(664704) parameters.  Although SRResNetNB has more parameters, our SRResNetNB model is still acceptable which can be efficiently trained with a single GPU.\n\n\\subsection{The Position of RelU}\n\n\nIn the residual branches, convolutional and ReLU layers are applied. The performances compared with the positions of ReLU layers (ReLU before\/after conv) as in Fig.~\\ref{fig:mmm_Comp} are represented in Table~\\ref{table:mmm_ablations} on Set14. The compared network has a same depth and corresponding convolutional layers among these networks have the same parameter numbers.\n\n\\begin{figure}[thb]\n  \\centering\n  \\footnotesize\n  \\begin{tabular}{cc}\n\n  \\subfigure[ReLU before convolution]{\\includegraphics[width=0.220\\textwidth]{.\/fig\/Pre_act.pdf}} &\n    \\subfigure[ReLU after convolution]{\\includegraphics[width=0.220\\textwidth]{.\/fig\/after_act.pdf}} \\\\\n   \n  \\end{tabular}\n  \\caption{The positions of ReLU in residual branches.}\n  \\label{fig:mmm_Comp}\n\\end{figure}\n\n\\begin{table}[thb]\n  \\centering\n \n \n  \\caption{Ablation comparisons for residual network with different orders of convolution and ReLU layers in terms of average PSNR (dB) on Set14.}\n  \\label{table:mmm_ablations}\n  \\begin{tabular}{c|c|c}\n    \\hline\n      scale &identity$+$&identity$+$\\\\\n\t\t   &ReLU after conv&ReLU before conv\\\\\n    \\hline\n      $\\times$ 2 &   32.97    & 33.01\\\\\n      $\\times$ 3 &     29.75   & 29.77  \\\\\n      $\\times$ 4 &    28.02   & 28.02  \\\\\n    \\hline\n \\end{tabular}\n\\end{table}\n\nFrom the results in Table~\\ref{table:mmm_ablations}, we conclude that the positions of ReLU in the residual branches make small differences.\n\n\\subsection{Impacts of Batch Normalization on SISR}\n\nIn Fig.~\\ref{fig:Comp_b}, test PSNR of Set 14 against training epochs by our R-basic with and without BN are compared to demonstrate the impacts of Batch Normalization on SISR problems.\nIn Fig.~\\ref{fig:Comp_b}, the compared structure with BN layers is the same as the structure applied for image classifications~\\cite{he2016identity}, showed in the rightmost column in Fig.~\\ref{fig:res-unit}.\n\nIt seems adding BN operations has hampered further improvement when more epoches have been performed. Normalizing input distribution of mini-batch to suppress data shifting has been proved powerful and largely accelerated the training convergency speed. It also enables deeper architectures and larger learning rates to be utilized in other tasks. However, whiten input and output of the intermediate layer may not be suitable for image super-resolution task which needs precise output. Another suspect may be regularization effects of BN have not been fully exploited as the training set of Fig.~\\ref{fig:Comp_b} is still limited in contrast with ImageNet. As larger learning rates were enabled by gradient clipping methods, the benefits of BN for leaning rates are alleviated.\n\n\\begin{figure}[!hbt]\n  \\centering\n  \\footnotesize\n    \\includegraphics[width=0.40\\textwidth]{.\/fig\/bn.pdf}\n  \\caption{The impacts of BN: test psnr of Set 14 against training epochs by our R-basic with and without BN.}\n  \\label{fig:Comp_b}\n\\end{figure}\n    \n\nFrom the perspective of centering activations, the skip connection itself has the benefits of centering the activations which partially reduces the necessities of BN operations when the network is not too deep to correct the mean bias. Moreover, the BN operation takes extra computations during learning and inference. Without BN operation, provided with certain computational resources, larger and wider deep architectures can be enabled to get better performances.\n\nThe impacts of Batch Normalization on SISR are still an open issue for the future study.\n\n\n\n\n\\subsection{The Deeper the Better, the Wider the Better}\n\nSRResNetNB performs much better than R-basic and VDSR model with deeper and wider network. Next, ablations of our system would be evaluated to unpack this performance gain. The skip connections and two factors, width (related to filter numbers) and depth of our model would be analyzed in the following steps.\n\n\\begin{table}[thb]\n       \\centering\n           \n          \n           \n           \n            \\caption{PSNR comparison between our residual CNN and {VDSR} trained by us}\n           \\label{table:VDSR-R}\n            \\begin{tabular}{c|c|c|c|c}\n            \\hline\n            &Set5&Set14&BSD100&Urban100\\\\\n             \\hline\n             \\hline\n            $R(64_8)$&37.28&32.91&31.72&30.45\\\\\n            \\hline\n            VDSR&37.32&32.89&31.77&30.51\\\\\n             \\hline\n            \\end{tabular}\n\\end{table}\n\n\nFirst, 20-layer VDSR has been added with 8 identity shortcuts to form a residual network, denoted $R(64_8)$. Each residual unit has two convolutional and Relu layers. The performance of $R(64_8)$ is roughly the same as VDSR in Table~\\ref{table:VDSR-R}. The shortcuts have very little impacts on the descriptive power. From the perspective of the centering the activations, to predict the high frequencies of image has pushed the final output activation centered. Within certain depth (\\textit{e}.\\textit{g}.,~20), the difficulties of learning has been alleviated, thus the shortcuts of the residual network have less impacts on the descriptive power.\n\\begin{table*}[!hbt]\n       \\centering\n          \n           \n           \n            \\caption{PSNR by the residual model of different depth and width with a magnification factors 2 in Set14.}\n           \\label{table:dw}\n            \\begin{tabular}{c|c|c|c|c|c}\n            \\hline\n           & $R(16_3,32_3,64_3)$ &$R(32_3,64_3,128_3)$ &$R(16_3,32_3,64_3,128_3)$& $R(16_3,32_3,64_3,128_3,256_3)$ &$R(4_3,8_3,16_3,32_3,64_3)$\\\\\n           \\hline\n           \\hline\n           PSNR(dB)&32.85& 32.96 & 33.00 &33.10&32.91\\\\\n             \\hline\n            \\end{tabular}\n\\end{table*}\n\nIf the network goes even deeper, the mean bias accumulate and difficulties of training increase. Then the benefits of skip connection will dominate that it alleviates gradient vanishing\/exploding problems and helps centering the activations in the layers of the net, which enable a deeper network and greatly improve the performance.\n\n\nSecond, fixing the depth of the model, simply broadening the width will improve the performance as showed in Table~\\ref{table:dw}, \\textit{e}.\\textit{g}., $R(16_3,32_3,64_3)$ vs $R(32_3,64_3,128_3)$, $R(4_3,8_3,16_3,32_3,64_3)$ vs $R(16_3,32_3,64_3,128_3,256_3)$. Increasing the filter numbers would enlarge the model capacity which enables modeling more complex nonlinear mappings from LR examples to HR examples.\n\nThird, the deeper the architecture, the better the performance. Adding more residual units, \\textit{e}.\\textit{g}., $R(16_3,32_3,64_3,128_3)$ vs $R(32_3,64_3,128_3)$ will improve the performance. Certainly, the depth should be no more than certain limit to avoid the overfitting problem and computational resource limitations. Within this limit, the deeper the better. Our residual unit eases the training difficulties which enables a deeper CNN architecture to improve the situation. On the other side, when model goes deeper as our residual SRResNetNB, plain deep CNN like VDSR fashions can not converge well and the restorations deteriorate. Another attempt to facilitate deeper net is the lightweight design which aims to solve the problem of too many parameters. It will be discussed next.\n\n\n\\subsection{Lightweight Design}\n\nIn this part, the proposed strategy of gradually varying the `shape' of residual network has been investigated.  The performances of different architectures with different shapes have been investigated for our residual net in Table~\\ref{table:shape-R} and {VDSR} fashions in Table~\\ref{table:shape-V} counterpart.\n\nThe number of featuremap has been gradually varied. To be specific, there are 28 layers as 6 \\textbf{container}s stack, each \\textbf{container} contains 2 residual units (2 convolutional layers in each residual unit). The depth can be calculated as $28=2+6\\times2\\times2+2$, where feature representation sub-network and reconstruction sub-network each have 2 convolutional layers. For models of VDSR fashions, 12-layer VDSR have been explored. For residual architectures, networks of different `shapes' have achieved comparable results. On the contrary, the performances of VDSR structures have largely fluctuated when the shapes of the networks vary.\n\\begin{table*}[thb]\n\\centering\n            \n          \n           \n\\caption{Performance by different residual models which have different shapes with a magnification factor 2 in Set14.}\n\\label{table:shape-R}\n\\scriptsize\n\\begin{tabular}{c|c|c|c|c}\n            \\hline\n           residual& $R(16_4,32_4,64_4)$& $R(64_4,32_4,16_4)$ & $R(16_2,32_2,64_2,64_2,32_2,16_2)$& $R(64_2,32_2,16_2,16_2,32_2,64_2)$\\\\%&  $R(64_6)$\\\\\n            \\hline\n             \\hline\n            PSNR(dB)& 32.91&32.85&32.94&32.89\\\\%&32.73\\\\\n            \\hline\n\\end{tabular}\n\\end{table*}\n\n\\begin{table*}[thb]\n\\centering\n            \n          \n           \n\\caption{Performance by {VDSR} models which have different shapes with a magnification factor 2 in Set14.}\n\\label{table:shape-V}\n\\begin{tabular}{c|c|c|c|c|c}\n            \\hline\n            {VDSR}& $(8_2,16_2,64_2)$ &$(64_2,16_2,8_2)$ & $(8,16,64,64,16,8)$&$(64,16,8,8,16,64)$& $(64_{16})$\\\\\n            \\hline\n            \\hline\n              PSNR(dB)&32.68&32.59&32.66 &32.50&32.85\\\\\n            \\hline\n\\end{tabular}\n\\end{table*}\n \n\nResidual networks can be interpreted as an ensemble of models which are the paths of differing depth in the residual network~\\cite{Veit2016Residual}. When the `shapes' of residual models are gradually changing, some short paths in the residual network have been less disturbed as Fig.~\\ref{fig:ensemble}. Thus, the performances of the ensembles are nearly unchanged.  On the contrary, the single path VDSR network are more disturbed by the variations of the shape. Instead of keeping the filter number fixed, less parameters can be applied for the residual network with our strategy to achieve comparable performances.\n\n\n\n\n\n\\subsection{Training with Multiple vs Single Upscaling Factors}\n\nIn this section, we compare the performances of networks handling multiple with respect to single upscaling factors as~\\cite{kim2016accurate} in Table~\\ref{table:Multi2single}. The training examples from different upscaling factors were mixed together to enable the model handling multiple upscaling factors. It seems mixing samples augmentations strategy~\\cite{kim2016accurate} from different upscaling factors has slightly boosted the performances, especially for large upscaling factors.\n\n\\begin{table}[thb]\n       \\centering\n       \\setlength{\\tabcolsep}{3.5pt}\n          \n           \n           \n            \\caption{PSNR comparisons between models handling multiple vs single upscaling factors, denoted as `Multiscale' and `single scale'}\n           \\label{table:Multi2single}\n            \\begin{tabular}{c|c|c|c|c|c}\n            \\hline\n                 \\multicolumn{2}{c}{} &Set5&Set14&BSD100&Urban100\\\\\n             \\hline\n                   \\multicolumn{2}{c}{} & PSNR\/SSIM & PSNR\/SSIM & PSNR\/SSIM & PSNR\/SSIM \\\\\n             \\hline\n           \\multirow{2}{*}{$\\times$ 2}& Multiscale& 37.51\/0.9587&  \\textbf{33.10}\/\\textbf{0.9131}&\\textbf{31.91}\/\\textbf{0.8961} & \\textbf{30.88}\/\\textbf{0.9150}\\\\\n\n            &single scale& \\textbf{37.52}\/\\textbf{0.9589}&33.03\/0.9129&31.90\/0.8958&30.84\/0.9143\\\\\n             \\hline\n             \\hline\n               \\multirow{2}{*}{$\\times$ 3}& Multiscale& \\textbf{33.72}\/\\textbf{0.9215}& \\textbf{29.80}\/\\textbf{0.8317} &\\textbf{28.83}\/\\textbf{0.7980} & \\textbf{27.17}\/\\textbf{0.8283}\\\\\n\n            &single scale&33.6\/0.9212 & 29.75\/0.8313&28.79\/0.7967& 27.08\/0.8255\\\\\n             \\hline\n              \\hline\n               \\multirow{2}{*}{$\\times$ 4}& Multiscale&\\textbf{31.37}\/\\textbf{0.8838} &\\textbf{28.06}\/\\textbf{0.7681}  &\\textbf{27.29}\/\\textbf{0.7251} &\\textbf{25.22}\/\\textbf{0.7537} \\\\\n\n            &single scale&31.30\/0.8824& 27.99\/0.7668&27.24\/0.7237& 25.14\/0.7051\\\\\n             \\hline\n            \\end{tabular}\n\\end{table}\n\n\n\n\n\n\n\\section{conclusion}\n\nIn this paper, from the perspective of centering activations and ensemble behaviors of residual network, a novel residual deep CNN which takes advantage of skip connections or identity mapping shortcuts in avoiding gradient exploding\/vanishing problem was proposed for single image super-resolution. In particular, the `shape' of CNN has been carefully designed such that a very deep convolutional neural network with much fewer parameters can produce even better performance. Based on the investigations into the influences of the network `shape' on the performances, a strategy of gradually varying the `shape' of the network has been proposed to construct this lightweight model.\nExperimental results have demonstrated that the proposed method can not only achieve state-of-the-art PSNR and SSIM results for single image super-resolution but also produce visually pleasant results.\n\n\n\\begin{figure*}[th]\n  \\centering\n  \\setlength{\\tabcolsep}{2pt}\n  \\footnotesize\n  \\begin{tabular}{p{1cm}cccc}\n   \n    $\\bigotimes{2}$&\n     \\subfigure{\\includegraphics[width=0.20\\textwidth]{.\/pic\/SRCNN_2x2.pdf}}&\n    \\subfigure{\\includegraphics[width=0.20\\textwidth]{.\/pic\/RFL_2x2.pdf}} &\n    \\subfigure{\\includegraphics[width=0.20\\textwidth]{.\/pic\/VDSR_2x2.pdf}}&\n      \\subfigure{\\includegraphics[width=0.20\\textwidth]{.\/pic\/vresidual_2x2.pdf}} \\\\\n     &27.35db &27.24db& 28.61db&29.01db\\\\\n       $\\bigotimes{3}$ &\n       \\subfigure{\\includegraphics[width=0.20\\textwidth]{.\/pic\/SRCNN_2x3.pdf}}&\n    \\subfigure{\\includegraphics[width=0.20\\textwidth]{.\/pic\/RFL_2x3.pdf}} &\n    \\subfigure{\\includegraphics[width=0.20\\textwidth]{.\/pic\/VDSR_2x3.pdf}}&\n      \\subfigure{\\includegraphics[width=0.20\\textwidth]{.\/pic\/vresidual_2x3.pdf}} \\\\\n     &34.93db& 35.20db &  36.67db & 36.91db\\\\\n       $\\bigotimes{4}$ &\n       \\subfigure{\\includegraphics[width=0.20\\textwidth]{.\/pic\/SRCNN_2x4.pdf}}&\n    \\subfigure{\\includegraphics[width=0.20\\textwidth]{.\/pic\/RFL_2x4.pdf}} &\n    \\subfigure{\\includegraphics[width=0.20\\textwidth]{.\/pic\/VDSR_2x4.pdf}}&\n      \\subfigure{\\includegraphics[width=0.20\\textwidth]{.\/pic\/vresidual_2x4.pdf}} \\\\\n   & 25.70db & 25.72db &25.79db  &26.00db\\\\\n    upscaling factors &\\textbf{SRCNN}\\cite{dong2014learning} & \\textbf{RFL}\\cite{schulter2015fast} & \\textbf{VDSR}\\cite{kim2016accurate} & \\textbf{ours}\\\\\n    GT&\n    \\subfigure{\\includegraphics[width=0.20\\textwidth]{.\/pic\/GT_2x2.pdf}}&\n    \\subfigure{\\includegraphics[width=0.20\\textwidth]{.\/pic\/GT_2x3.pdf}} &\n    \\subfigure{\\includegraphics[width=0.20\\textwidth]{.\/pic\/GT_2x4.pdf}}&\\\\\n  \\end{tabular}\n  \\caption{Comparisons of image SR results with different methods in different upscaling factors}\n  \\label{fig:SRresults}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgment}\n\n This work is partially supported by National Science Foundation of China under\nGrant NO. 61473219.\n\n\n{\\small\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe theory of complete minimal surfaces in $\\mathbb{H}^2\\times\\mathbb{R}$ with finite total curvature, i.e., those whose Gauss curvature is integrable, has received considerable attention during the last decade, mainly triggered by Collin and Rosenberg~\\cite{CR}. The combined work of Hauswirth, Nelli, Sa Earp and Toubiana~\\cite{HNST}, and Hauswirth, Menezes and Rodr\u00edguez~\\cite{HMR} shows that a complete minimal surface immersed in $\\mathbb{H}^2\\times\\mathbb{R}$ has finite total curvature if and only if it is proper, has finite topology and each of its ends is asymptotic to an admissible polygon, i.e., a curve homeomorphic to $\\mathbb{S}^1$ consisting of finitely-many alternating complete vertical and horizontal ideal geodesics, see~\\cite{HMR}. Here the product compactification of $\\mathbb{H}^2\\times\\mathbb{R}$ is considered, in which the horizontal (resp.\\ vertical) ideal boundary consists of two disks $\\mathbb H^2\\times\\{\\pm\\infty\\}$ (resp.\\ the cylinder $\\partial_{\\infty}\\mathbb H^2\\times\\mathbb{R}$). Ideal horizontal geodesics are those of the form $\\Gamma\\times\\{+\\infty\\}$ or $\\Gamma\\times\\{-\\infty\\}$, being $\\Gamma$ a geodesic of $\\mathbb H^2$, whereas ideal vertical geodesics are those of the form $\\{p_\\infty\\}\\times\\mathbb{R}$, where $p_\\infty\\in\\partial_\\infty\\mathbb H^2$ is an ideal point.\n\nCombining the above classification with the previous work of Hauswirth and Rosenberg~\\cite{HR}, the following Gauss--Bonnet-type formula for a complete minimal surface $\\Sigma$ immersed in $\\mathbb{H}^2\\times\\mathbb{R}$ with finite total curvature holds true:\n\\begin{equation}\\label{eqn:generalized-GB}\n\\int_\\Sigma K=2\\pi\\chi(\\Sigma)-2\\pi m=2\\pi(2-2g-k-m),\n\\end{equation}\nwhere $g$ and $k$ are the genus and the number of ends of $\\Sigma$, respectively, $\\chi(\\Sigma)=2-2g-k$ its Euler characteristic, $K$ its Gauss curvature, and $m$ is the total number of horizontal ideal geodesics in $\\mathbb{H}^2\\times\\{+\\infty\\}$, among all polygonal components associated with the ends of $\\Sigma$. Observe that the union of all these components consists of $m$ ideal horizontal geodesics in $\\mathbb{H}^2\\times\\{+\\infty\\}$, $m$ ideal horizontal geodesics in $\\mathbb{H}^2\\times\\{-\\infty\\}$, and $2m$ ideal vertical geodesics, so the term $2\\pi m$ in~\\eqref{eqn:generalized-GB} can be understood as the sum of exterior angles of the asymptotic boundary of $\\Sigma$. We also remark that Formula~\\eqref{eqn:generalized-GB} has been extended to some quotients of $\\mathbb{H}^2\\times\\mathbb{R}$ by Hauswirth and Menezes~\\cite{HM}.\n\n\nAlthough this characterization is very satisfactory from a theoretical point of view, it seems tough in general to determine whether or not a given family of admissible polygons actually bounds a minimal surface, or if a given topological type can be realized by such a surface. In fact, there are not many examples of surfaces with finite total curvature in the literature. Let us highlight some of them in terms of the three parameters $(g,k,m)$ appearing in~\\eqref{eqn:generalized-GB}:\n\\begin{itemize}\n\\item  The simplest case is that of flat minimal surfaces, which must be vertical planes (i.e., of the form $\\Gamma\\times\\mathbb{R}$, being $\\Gamma\\subset\\mathbb{H}^2$ a complete geodesic) because of Gauss equation. In particular, vertical planes are the only complete minimal surfaces with finite total curvature and  $(g,k,m)=(0,1,1)$, see also~\\cite[Corollary~5]{HST}. \n\\item A minimal Scherk graph in $\\mathbb{H}^2\\times\\mathbb{R}$ is a minimal graph over a geodesic ideal polygon of $\\mathbb{H}^2$ with $2a$ vertexes, $a\\geq 2$, taking alternating limit values $+\\infty$ and $-\\infty$ on the sides of the polygon. A characterization of polygons carrying such a surface is analyzed in~\\cite{MRR}, in which case they have finite total curvature and satisfy $(g,k,m)=(0,1,a)$. The case $a=2$ gives rise to the only complete minimal surfaces with total curvature $-2\\pi$, as shown by Pyo and Rodr\u00edguez~\\cite[Theorem~4.1]{PR}. We can find as well the \\emph{twisted} Scherk minimal surfaces~\\cite{PR} with $(g,k,m)=(0,1,2b+1)$, $b\\geq 1$, and total curvature $-4b\\pi$ that are no longer graphs or bigraphs, some of which are embedded.\n\\item Minimal $k$-noids constructed by Morabito and Rodr\u00edguez~\\cite{MorRod} (also by Pyo~\\cite{Pyo} in the symmetric case) have finite total curvature, genus $0$ and $k$ ends asymptotic to vertical planes. This gives $g=0$ and $k=m\\geq 2$. \n\\item Horizontal catenoids are the only complete minimal surfaces immersed in $\\mathbb{H}^2\\times\\mathbb{R}$ with finite total curvature and $k=m=2$, see~\\cite{HMR,HNST}. The family of minimal surfaces with finite total curvature and $k=m\\geq 3$ is not hitherto well understood, not even in the case $g=0$. The most general construction was given by Mart\u00edn, Mazzeo and Rodr\u00edguez~\\cite{MMR}, who found properly embedded minimal surfaces with finite total curvature in $\\mathbb{H}^2\\times\\mathbb{R}$ of genus $g$ and $k$ ends asymptotic to vertical planes (and hence $m=k$), for arbitrary $g\\geq 0$ and $k$ arbitrarily large depending on $g$.\n\\end{itemize}\n\nIn this paper we provide highly symmetric examples with $g=1$ and $m=k\\geq 3$, which are hence conformally equivalent to a torus with $k$ punctures. They can be thought of as the counterpart in $\\mathbb{H}^2\\times\\mathbb{R}$ of the genus $1$ minimal $k$-noids in $\\mathbb{R}^3$ obtained by Mazet~\\cite{Maz}. Outside a compact subset, our surfaces look like the minimal $k$-noids in~\\cite{MorRod,Pyo}, and they are not globally embedded in general. Notice that there are no such examples with $k=2$ due to the aforesaid uniqueness of horizontal catenoids in~\\cite{HNST}. Our main result can be stated as follows:\n\n\\begin{theorem}~\\label{thm:knoids}\nFor each $k\\geq 3$, there exists a $1$-parameter family of properly Alexandrov-embedded minimal surfaces in $\\mathbb{H}^2\\times\\mathbb{R}$ with genus $1$ and $k$ ends, dihedrally symmetric with respect to $k$ vertical planes and symmetric with respect to a horizontal plane. They have finite total curvature $-4k\\pi$ and each of their ends is embedded and asymptotic to a vertical plane. \n\\end{theorem}\n\nThe construction of these genus $1$ minimal $k$-noids is based on a conjugate technique, in the sense of Daniel~\\cite{Dan} and Hauswirth, Sa Earp and Toubiana~\\cite{HST}. Conjugation has been a fruitful technique to obtain constant mean curvature surfaces in $\\mathbb{H}^2\\times\\mathbb{R}$ and $\\mathbb{S}^2\\times\\mathbb{R}$, see~\\cite{ManTor,ManTor2,MPT,Maz,MRR2,MorRod,Plehnert,Plehnert2,Pyo} and the references therein. We begin by considering a solution to an improper Dirichlet problem~\\cite{MRR,NR} in $\\mathbb{H}^2\\times\\mathbb{R}$ over an unbounded geodesic triangle $\\Delta\\subset\\mathbb{H}^2$, a so-called Jenkins--Serrin problem~\\cite{MRR}. These solutions are minimal graphs over the interior of $\\Delta$ with prescribed finite and infinite values when one approaches $\\partial\\Delta$. The conjugate surface is another minimal graph in $\\mathbb{H}^2\\times\\mathbb{R}$ whose boundary is made of curves lying on totally geodesic surfaces, i.e., vertical and horizontal planes. Since there are isometric reflections across such planes in $\\mathbb{H}^2\\times\\mathbb{R}$, the conjugate surface can be extended to a complete surface under suitable conditions. In order to prescribe the symmetries stated in Theorem~\\ref{thm:knoids}, we will encounter two period problems that will impose further restrictions on $\\Delta$ and on the boundary values of the Jenkins--Serrin problem. \n\nOur conjugate approach is inspired by the genus $1$ minimal $k$-noids in $\\mathbb{R}^3$ given by Mazet~\\cite{Maz}, and by the mean curvature $\\frac{1}{2}$ surfaces in $\\mathbb{H}^2\\times\\mathbb{R}$ given by Plehnert~\\cite{Plehnert}. It is important to remark that there exist technical dissimilarities between the cases $H=0$ and $H=\\frac{1}{2}$ in $\\mathbb{H}^2\\times\\mathbb{R}$ because of the fact that the conjugate of a surface with mean curvature $\\frac{1}{2}$ (resp.\\ $0$) is a minimal surface in Heisenberg group $\\mathrm{Nil}_3$ (resp.\\ $\\mathbb{H}^2\\times\\mathbb{R}$). Furthermore, our construction can be adapted to produce complete minimal surfaces invariant by an arbitrary vertical translation (i.e., in the direction of the factor $\\mathbb{R}$), similar to the saddle towers given in~\\cite{MorRod}. They have genus $1$ in the quotient and they are not embedded in general.\n\n\\begin{theorem}\\label{thm:saddle-towers}\nFor each $k\\geq 3$ and each vertical translation $T$, there is a $1$-parameter family of Alexandrov-embedded singly periodic minimal surfaces in $\\mathbb{H}^2\\times\\mathbb{R}$ invariant by $T$ and dihedrally symmetric with respect to $k$ vertical planes and a horizontal plane. They have finite total curvature $-4k\\pi$, genus $1$ and $2k$ vertical ends in the quotient of $\\mathbb{H}^2\\times\\mathbb{R}$ by $T$.\n\\end{theorem}\n\nOur analysis of the period problems will allow us to find surfaces that are not invariant by a discrete group of rotations, but by discrete groups of parabolic or hyperbolic translations, which we will call \\emph{parabolic and hyperbolic $\\infty$-noids}, respectively. These surfaces have infinitely many ends, and we can guarantee that many of the examples are properly embedded in the hyperbolic case. Although we will not state it explicitly, analogous surfaces can be obtained in the quotient by an arbitrary vertical translation in the spirit of Theorem~\\ref{thm:saddle-towers}.\n\n\\begin{theorem}\\label{thm:infty-noids}\nThere is a $2$-parameter (resp. $1$-parameter) family of properly embedded (resp. Alexandrov-embedded) minimal surfaces in $\\mathbb{H}^2\\times\\mathbb{R}$ with genus $0$ and infinitely many ends, invariant by a discrete group of hyperbolic (resp. parabolic) translations. Each of their ends is embedded, asymptotic to a vertical plane, and has finite total curvature.\n\\end{theorem}\n\nThe paper is organized as follows: In Section~\\ref{sec:preliminaries} we will analyze some aspects of the conjugation of surfaces in $\\mathbb{H}^2\\times\\mathbb{R}$ that will be needed in the construction, and Section~\\ref{sec:construction} will be devoted to fill the details of the proof of Theorems~\\ref{thm:knoids} and~\\ref{thm:saddle-towers}. We will also discuss some open questions about the uniqueness and embeddedness of the constructed surfaces, as well as natural limits of the $1$-parameter family of genus $1$ $k$-noids. In the last part of the paper we will prove Theorem~\\ref{thm:infty-noids}.\n\n\\medskip\n\\noindent\\textbf{Acknowledgment.} The authors would like to express their gratitude to Magdalena Rodr\u00edguez for her valuable comments during the preparation of this manuscript, as well as the anonymous referee for their thorough revision of the manuscript, which has greatly improved the final presentation. This research was supported by \\textsc{mineco--feder} project MTM2017-89677-P. The first author is also supported by the FPU program from \\textsc{micinn}. The second author is also supported by the \\textsc{micinn--feder} project PID2019-111531GA-I00\/AEI\/10.13039\/501100011033.\n\n\n\\section{Preliminaries}\\label{sec:preliminaries}\n\n\nLet $\\Sigma$ be a simply connected Riemannian surface. Given an isometric minimal immersion $X:\\Sigma\\to\\mathbb{H}^2\\times\\mathbb{R}$, Hauswirth, Sa Earp and Toubiana~\\cite{HST} proved the existence of another isometric minimal immersion $\\widetilde X:\\Sigma\\to\\mathbb{H}^2\\times\\mathbb{R}$ such that:\n\\begin{enumerate}\n\t\\item Both immersions induce the same angle function $\\nu=\\langle N,\\partial_t\\rangle=\\langle\\widetilde N,\\partial_t\\rangle$, where $N$ and $\\widetilde N$ stand for unit normal vector fields to $X$ and $\\widetilde X$, respectively, and $\\partial_t$ is the unit vector field in $\\mathbb{H}^2\\times\\mathbb{R}$ in the direction of the factor $\\mathbb{R}$.\n\t\\item The shape operators $S$ and $\\widetilde S$ of $X$ and $\\widetilde X$, respectively, satisfy $\\widetilde S=JS$, where $J$ is the $\\frac\\pi2$-rotation in $T\\Sigma$, chosen such that both $\\{\\mathrm{d}X_p(u),\\mathrm{d}X_p(Ju),N_p\\}$ and $\\{\\mathrm{d}\\widetilde X_p(u),\\mathrm{d}\\widetilde X_p(Ju),\\widetilde N_p\\}$ are positively oriented bases in $\\mathbb H^2\\times\\mathbb R$ for all non-zero tangent vectors $u\\in T_p\\Sigma$.\n\t\\item The tangential components $T=\\partial_t-\\nu N$ and $\\widetilde T=\\partial_t-\\nu\\widetilde N$ of $\\partial_t$ satisfy $\\widetilde X^*\\widetilde T=JX^*T$. This implies that $\\langle\\mathrm{d}X_p(u),\\partial_t\\rangle=\\langle\\mathrm{d}\\widetilde X_p(Ju),\\partial_t\\rangle$ for all $u\\in T_p\\Sigma$.\n\\end{enumerate}\nThe immersions $X$ and $\\widetilde X$ are called \\emph{conjugate} and determine each other up ambient isometries preserving both the global orientation and the vector field $\\partial_t$. Our initial surface $X(\\Sigma)$ will be a vertical graph over a convex domain, namely a solution of a Jenkins--Serrin problem. This implies that $\\widetilde X(\\Sigma)$ is also a vertical graph over another (possibly non-convex) domain, due to the Krust-type theorem given by~\\cite[Theorem~14]{HST}. Therefore, we can assume that both surfaces are embedded and will use the notation $\\Sigma$ and $\\widetilde\\Sigma$ for the surfaces $X(\\Sigma)$ and $\\widetilde X(\\Sigma)$, respectively.\n\nAlthough the conjugate surface $\\widetilde\\Sigma$ is not explicit in general, one can obtain insightful information if the initial surface $\\Sigma$ has boundary consisting of horizontal and vertical geodesics intersecting at some vertexes. A curve $\\Gamma\\subset\\Sigma$ is a horizontal (resp.\\ vertical) geodesic in $\\mathbb{H}^2\\times\\mathbb{R}$ if and only if the conjugate curve $\\widetilde\\Gamma\\subset\\widetilde\\Sigma$ lies in a vertical (resp.\\ horizontal) totally geodesic surface of $\\mathbb{H}^2\\times\\mathbb{R}$ intersecting $\\widetilde\\Sigma$ orthogonally along $\\widetilde\\Gamma$. Furthermore, axial symmetry about $\\Gamma$ corresponds to mirror symmetry about $\\widetilde\\Gamma$, which enables analytic continuation of $\\Sigma$ and $\\widetilde\\Sigma$ across their boundaries. If the angles at the vertexes of $\\partial\\Sigma$ are integer divisors of $\\pi$, then no singularity appears at such vertexes after successive reflections about the boundary components, and both surfaces can be extended to complete (possibly non-embedded) minimal surfaces. We refer to~\\cite{ManTor,Plehnert,MRR2} for details.\n\nHowever, most difficulties concerning the depiction of $\\widetilde\\Sigma$, and in particular deciding whether or not it is embedded, show up when one tries to understand the behavior of the conjugate of a vertical geodesic. We will now recall some properties on this matter which will be used later in Section~\\ref{sec:construction}. Let $\\gamma:I\\to\\partial\\Sigma$ be a vertical geodesic with unit speed such that $\\gamma'=\\partial_t$ (this orientation of vertical geodesics will be fixed throughout the text), where $I\\subset\\mathbb{R}$ is an interval, and denote by $\\widetilde\\gamma:I\\to\\partial\\widetilde\\Sigma$ the conjugate curve, which will be assumed to lie in $\\mathbb{H}^2\\times\\{0\\}$ after a vertical translation. \n\nLet us consider the half-space model $\\mathbb{H}^2\\times\\mathbb{R}=\\{(x,y,t)\\in\\mathbb{R}^3:y>0\\}$, whose metric is given by $y^{-2}(\\mathrm{d}x^2+\\mathrm{d}y^2)+\\mathrm{d}t^2$, with positively oriented orthonormal frame $\\{E_1,E_2,\\partial_t\\}$ given by $E_1=y\\partial_x$ and $E_2=y\\partial_y$ (observe that $E_1$ is tangent to the foliation of $\\mathbb{H}^2$ by horocycles $y=y_0$ with $y_0>0$). Since $\\gamma$ is vertical and $\\widetilde\\gamma$ lies in a horizontal slice, there exist smooth functions $\\psi,\\theta\\in C^\\infty(I)$ such that\n\\begin{align}\n N_{\\gamma(t)}&=\\cos(\\psi(t))E_1+\\sin(\\psi(t))E_2,\\label{eqn:rotation-angle}\\\\\n \\widetilde\\gamma'(t)&=\\cos(\\theta(t))E_1+\\sin(\\theta(t))E_2,\\label{eqn:foliation-angle}\n\\end{align}\ncalled the angle of rotation of $N$ along $\\gamma$ and the angle of rotation of $\\widetilde\\gamma$ with respect to the foliation by horocycles, respectively. We now collect some relations between these quantities, see also~\\cite{CMR,MPT,Plehnert}.\n\nObserve that $E_1$ and $E_2$ are parallel vector fields along $\\gamma$ since they satisfy $\\overline\\nabla_{\\partial_t}E_1=\\overline\\nabla_{\\partial_t}E_2=0$, where $\\overline\\nabla$ stands for the ambient Levi--Civita connection. This is due to the fact that that $\\mathbb H^2\\times\\mathbb{R}$ is a Riemannian product and $E_1$ and $E_2$ do not depend on the variable $t$. By taking derivatives in~\\eqref{eqn:rotation-angle}, we get $\\overline\\nabla_{\\gamma'}N=-\\psi'\\sin(\\psi)E_1+\\psi'\\cos(\\psi)E_2=-\\psi' N\\times\\gamma'$, where $\\times$ is the cross-product in $\\mathbb H^2\\times\\mathbb{R}$. Using the properties of the conjugation, we deduce the identity\n\\begin{equation}\\label{eqn:kg}\n\\psi'=-\\langle\\overline\\nabla_{\\gamma'}N,N\\times\\gamma'\\rangle=\\langle S\\gamma',J\\gamma'\\rangle=-\\langle J\\widetilde S\\widetilde\\gamma',J\\widetilde\\gamma'\\rangle=\\langle\\overline\\nabla_{\\widetilde\\gamma'}\\widetilde N,\\widetilde\\gamma'\\rangle=-\\kappa_g,\n\\end{equation}\nwhere $\\kappa_g$ is the geodesic curvature of $\\widetilde\\gamma$ as a curve of $\\mathbb{H}^2\\times\\{0\\}$ with respect to the conormal $\\widetilde N$ (recall that $\\widetilde\\Sigma$ intersects $\\mathbb{H}^2\\times\\{0\\}$ orthogonally). Now we will obtain further information under the additional assumption that the surfaces are \\emph{multigraphs}, i.e., their common angle function $\\nu$ has a sign.\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{fig-frames.png}\n\\caption{Orientation of the conjugate surfaces $\\Sigma$ and $\\widetilde\\Sigma$ according to the direction of rotation of $N$ along a vertical geodesic $\\gamma$.}\\label{fig:orientation}\n\\end{center}\\end{figure}\n\n\\begin{lemma}\\label{lem:orientation}\n Assume that the interiors of $\\Sigma$ and $\\widetilde\\Sigma$ are multigraphs over (possibly immersed) domains $\\Omega$ and $\\widetilde\\Omega$, respectively, with angle function $\\nu>0$, and let $\\gamma$ be a vertical geodesic in $\\partial\\Sigma$ with $\\gamma'=\\partial_t$. In the above notation:\n\\begin{enumerate}[label=\\emph{(\\alph*)}]\n\t\\item If $\\psi'>0$, then $J\\gamma'$ (resp.\\ $J\\widetilde\\gamma'=\\partial_t$) is a unit outer conormal to $\\Sigma$ (resp.\\ $\\widetilde\\Sigma$) along $\\gamma$ (resp.\\ $\\widetilde\\gamma$), $\\widetilde N$ points to the interior of $\\widetilde\\Omega$ along $\\widetilde\\gamma$, and $\\widetilde\\Sigma$ lies in $\\mathbb{H}^2\\times(-\\infty,0]$ locally around $\\widetilde\\gamma$ (see Figure~\\ref{fig:orientation}, top). \n\t\\item If $\\psi'<0$, then $J\\gamma'$ (resp.\\ $J\\widetilde\\gamma'=\\partial_t$) is a unit inner conormal to $\\Sigma$ (resp.\\ $\\widetilde\\Sigma$) along $\\gamma$ (resp.\\ $\\widetilde\\gamma$), $\\widetilde N$ points to the exterior of $\\widetilde\\Omega$ along $\\widetilde\\gamma$, and $\\widetilde\\Sigma$ lies in $\\mathbb{H}^2\\times[0,+\\infty)$ locally around $\\widetilde\\gamma$ (see Figure~\\ref{fig:orientation}, bottom). \n\\end{enumerate}\nEither way, the identity $\\theta'=\\psi'-\\cos(\\theta)$ holds true.\n\\end{lemma}\n\n\\begin{proof}\nWe will only prove item (a) since item (b) is analogous, so we will suppose that $\\psi'>0$. As $\\{\\gamma',J\\gamma',N\\}$ is positively oriented and $\\nu>0$, it follows that $J\\gamma'$ points towards the exterior of $\\Sigma$ along $\\gamma$ (see Figure~\\ref{fig:orientation}, top left). Since the rotation $J$ is intrinsic, we deduce that $J\\widetilde\\gamma'$ points to the exterior of $\\widetilde\\Sigma$ along $\\widetilde\\gamma$, and $\\widetilde N=\\widetilde\\gamma'\\times J\\widetilde\\gamma'$ is determined by the ambient orientation.\n\nAssume now by contradiction that $\\widetilde N$ points to the exterior of $\\widetilde\\Omega$ at some point $p$ of $\\widetilde\\gamma$. Since $\\kappa_g=-\\psi'<0$ with respect to the conormal $\\widetilde N$ and $\\nu>0$, we infer that $\\widetilde\\Sigma$ projects locally into the convex side of $\\widetilde\\gamma$. This yields a contradiction with the boundary maximum principle by comparing $\\widetilde\\Sigma$ and a vertical plane tangent to $\\widetilde\\gamma$ at $p$. Note that $J\\widetilde\\gamma'$ cannot be equal to $-\\partial_t$ (so it must be $J\\widetilde\\gamma'=\\partial_t)$ because it points outside $\\widetilde\\Sigma$ along $\\widetilde\\gamma$ and the angle function is positive. As a consequence, a neighborhood of $\\widetilde\\gamma$ in $\\widetilde\\Sigma$ is contained in $\\mathbb{H}^2\\times(-\\infty,0]$.\n\nIt is easy to calculate $\\overline\\nabla_{E_1}E_1=E_2$, $\\overline\\nabla_{E_1}E_2=-E_1$ and $\\overline\\nabla_{E_2}E_1=\\overline\\nabla_{E_2}E_2=0$ by using the expressions of $E_1$ and $E_2$ and Koszul formula. On the one hand, this allows us to take derivatives in~\\eqref{eqn:foliation-angle} to obtain $\\overline\\nabla_{\\widetilde\\gamma'}\\widetilde\\gamma'=(\\theta'+\\cos(\\theta))(-\\sin(\\theta)E_1+\\cos(\\theta)E_2)$. On the other hand, the above discussion shows that $\\widetilde N=\\widetilde\\gamma'\\times J\\widetilde\\gamma'=\\widetilde\\gamma'\\times \\partial_t=\\sin(\\theta)E_1-\\cos(\\theta)E_2$, so the last identity in the statement follows from plugging these computations in the expression $-\\psi'=\\kappa_g=\\langle\\overline\\nabla_{\\widetilde\\gamma'}\\widetilde\\gamma',\\widetilde N\\rangle$.\n\\end{proof}\n\n\n\\section{Construction of genus $1$ saddle towers and $k$-noids}\\label{sec:construction}\n\nThe first part of this section is devoted to prove Theorems~\\ref{thm:knoids} and~\\ref{thm:saddle-towers}. The arguments leading to these results are based on a conjugate construction that depends on a parameter $0<l\\leq\\infty$ that will be fixed henceforth. The case $0<l<\\infty$ gives the saddle towers whose fundamental pieces lie in a slab of height $l$, whereas the case $l=\\infty$ gives rise to the $k$-noids. Although a limit argument for $l\\to\\infty$ would imply the latter (as in~\\cite{MorRod}) we will discuss both cases together. \n\n\n\\subsection{The conjugate construction}\\label{subsec:periods} \n\nLet $\\Delta$ a geodesic triangle with sides $\\ell_1$, $\\ell_2$ and $\\ell_3$, and opposite vertexes $p_1$, $p_2$ and $p_3$. Assume that $\\Delta$ is acute and the length of $\\ell_2$ is given by the aforesaid parameter $l\\in(0,\\infty]$, so that $p_1$ is ideal if $l=\\infty$. Therefore $\\Delta$ is determined by the length $a$ of $\\ell_1$ and by the angle $\\varphi$ at $p_2$. Given $b\\in\\mathbb{R}$, consider the unique solution $\\Sigma(a,\\varphi,b)$ to the Jenkins--Serrin problem over $\\Delta$ with boundary values $b$ along $\\ell_1$, $+\\infty$ along $\\ell_2$, and $0$ along $\\ell_3$. Existence and uniqueness of such a solution is guaranteed under these boundary conditions, see~\\cite{MRR} and the references therein. In particular, the interior of $\\Sigma(a,\\varphi,b)$ is a minimal graph over $\\Delta$ whose boundary consists of two horizontal geodesics $h_1$ and $h_3$ lying in $\\mathbb{H}^2\\times\\{b\\}$ and $\\mathbb{H}^2\\times\\{0\\}$, respectively, and three vertical geodesics $v_1$, $v_2$ and $v_3$ projecting onto $p_1$, $p_2$ and $p_3$, respectively. Note that $v_1$ is an ideal vertical geodesic provided that $l=\\infty$. The boundary of $\\Sigma(a,\\varphi,b)$ also contains an horizontal ideal geodesic $h_2\\subset\\mathbb{H}^2\\times\\{+\\infty\\}$ projecting onto $\\ell_2$. \n\nSince $\\Delta\\subset\\mathbb{H}^2$ is convex, the conjugate minimal surface $\\widetilde\\Sigma(a,\\varphi,b)\\subset\\mathbb{H}^2\\times\\mathbb{R}$ is a graph over some domain $\\widetilde\\Delta\\subset\\mathbb{H}^2$ due to the Krust-type theorem in~\\cite{HST}. The normal $N$ along $v_2$ or $v_3$ rotates counterclockwise, so theoretical conjugate curves $\\widetilde{v}_2$ and $\\widetilde{v}_3$ lie in horizontal planes and are convex towards the exterior of $\\widetilde\\Delta$ by Lemma~\\ref{lem:orientation}. This lemma also implies that $\\widetilde\\Sigma(a,\\varphi,b)$ lies locally below the horizontal slices containing $\\widetilde{v}_2$ and $\\widetilde{v}_3$. We will assume that $\\widetilde v_3\\subset\\mathbb H^2\\times\\{0\\}$ in the sequel. If we decompose $\\widetilde h_i=(\\beta_i,z_i)\\in\\mathbb H^2\\times\\mathbb{R}$, $i\\in\\{1,2,3\\}$, where $\\widetilde h_i$ is the conjugate of $h_i$, then it is easy to check that $\\|\\beta_i'\\|=|\\nu|$ and $|z_i'|=\\sqrt{1-\\nu^2}$, so points at which $\\nu$ takes the values $0$ or $\\pm 1$ will be the key to understand the behavior of $\\widetilde h_i$.\n\n\\begin{lemma}\\label{lem:angle01}\nThe angle function $\\nu$ of $\\Sigma(a,\\varphi,b)$ is zero precisely at $v_2\\cup v_3$ if $l=\\infty$ or $v_1\\cup v_2\\cup v_3$ if $l<\\infty$. Furthermore, there is exactly one point of $\\Sigma(a,\\varphi,b)$ with $\\nu=1$ and it belongs to $h_1$.\n\\end{lemma}\n\n\\begin{proof}\nThe interior of $\\Sigma(a,\\varphi,b)$ is a graph so there are no interior zeros of $\\nu$. Besides, zeros in the interior of $h_i$, $i\\in\\{1,2,3\\}$, would contradict the boundary maximum principle by comparing $\\Sigma(a,\\varphi,b)$ and the vertical plane $\\ell_i\\times\\mathbb{R}$.\n\nIf $\\nu(p)=1$ at some interior point $p$, then the intersection of $\\Sigma(a,\\varphi,b)$ and the horizontal slice containing $p$ is an equiangular set of curves, and two of these curves starting at $p$ must reach some $h_i$ or have a common endpoint at some $v_i$. Either way, we can find a region of $\\Sigma(a,\\varphi,b)$ bounded by horizontal curves, and a standard application of the maximum principle with respect to horizontal slices gives a contradiction. If $l=\\infty$ and the endpoints are in $v_1$, then the region enclosed by the two curves has an end of type 1 in the sense of~\\cite[Definition~4.14]{MMR}, and the general maximum principle~\\cite[Theorem~4.16]{MMR} applies in this case. By the boundary maximum principle there are no points with $\\nu=1$ at $h_2\\cup h_3$, but the normal $N$ rotates an angle of $\\pi$ along $h_1$ (see Figure~\\ref{fig:conjugate-surfaces}), which gives rise to at least one point in $h_1$ with $\\nu=1$. If there were more than one, then by intersecting the surface and the horizontal slice containing $h_1$, we would get another region bounded by horizontal curves, which gives a similar contradiction as in the above argument.\n\\end{proof}\n\nWe deduce that $\\widetilde h_3$ projects one-to-one into the factors $\\mathbb{H}^2$ and $\\mathbb{R}$, as well as $\\widetilde h_1$ into the factor $\\mathbb H^2$. However, the component of $\\widetilde h_1$ in the factor $\\mathbb R$ has a minimum at the unique point where $\\nu=1$. On the other hand, the conjugate curve $\\widetilde{h}_2$ can be thought of as an ideal vertical geodesic of length $l$ in $\\partial_\\infty\\mathbb{H}^2\\times\\mathbb{R}$. Since $\\Sigma(a,\\varphi,b)$ becomes vertical when one approaches the side $\\ell_2$, the length of the ideal vertical geodesic $\\widetilde h_2$ is equal to $l$, in particular $\\widetilde v_1\\subset\\mathbb H^2\\times\\{-l\\}$. If $l<\\infty$, then $N$ rotates clockwise along $v_1$, so the conjugate curve $\\widetilde v_1$ lies in a horizontal slice (which $\\Sigma(a,\\varphi,b)$ approaches from above) and is convex towards the exterior of $\\widetilde\\Delta$ by Lemma~\\ref{lem:orientation}. If $l=\\infty$, then $\\widetilde{v}_1$ is an ideal horizontal geodesic in $\\mathbb{H}^2\\times\\{-\\infty\\}$, see also~\\cite[Corollary~2.4]{CMR}. A depiction of the conjugate surfaces is given in Figure~\\ref{fig:conjugate-surfaces}.\n\n\\begin{figure} \n\t\\includegraphics[width=0.9\\textwidth]{fig-contour.pdf}\n\t\\caption{Conjugate surfaces $\\Sigma(a,\\varphi,b)$ and $\\widetilde{\\Sigma}(a,\\varphi,b)$ and their domains $\\Delta$ and $\\widetilde\\Delta$ in $\\mathbb H^2$ in the case $l=\\infty$. Dashed lines represent ideal geodesics, and white dots represent ideal vertexes. The arrows in $\\Sigma(a,\\varphi,b)$ represent the normal $N$ at the endpoints of $v_2$ and $v_3$, which rotates counterclockwise along both geodesics.  }\\label{fig:conjugate-surfaces}\n\\end{figure}\n\nWe would like $\\widetilde v_2$ and $\\widetilde v_3$ to be contained in the same horizontal plane, as well as the complete geodesics of $\\mathbb{H}^2$ containing $\\widetilde h_1$ and $\\widetilde h_3$ to intersect at an angle of $\\frac{\\pi}{k}$ (as shown in Figure~\\ref{fig:conjugate-surfaces}, bottom right). Such a configuration would lead to the desired construction of saddle towers and $k$-noids, so we need to find values of $(a,\\varphi,b)$ that solve the following period problems, inspired by the arguments in~\\cite{Plehnert}:\n\\begin{enumerate}\n\\item \\textbf{First period problem.} Let $\\mathcal{P}_1(a,\\varphi,b)$ be the difference of heights of the horizontal curves $\\widetilde v_2$ and $\\widetilde v_3$, i.e., the difference of heights of the endpoints of $\\widetilde h_1$. Parameterizing $\\widetilde{h}_1:[0,a]\\to\\mathbb{H}^2\\times\\mathbb{R}$ with $\\widetilde h_1(0)\\in\\widetilde v_2$, $\\widetilde h_1(a)\\in\\widetilde v_3$ and unit speed, by means of the properties of the conjugation, we can express\n\\begin{equation}\\label{eqn:p1}\n\\mathcal{P}_1(a,\\varphi,b)=\\int_{\\widetilde h_1}\\langle\\widetilde h_1',\\partial_t\\rangle=\\int_{h_1}\\langle \\eta,\\partial_t\\rangle.\n\\end{equation}\nwhere $\\eta=-Jh_1'$ is the unit inward conormal vector to $\\Sigma(a,\\varphi,b)$ along $h_1$.\n\n\\item \\textbf{Second period problem.} Let us work in the half-space model. After an ambient isometry, we can assume that $\\widetilde{h}_3$ lies in the vertical plane $x=0$ and $\\widetilde{v}_2:[0,b]\\to\\mathbb{H}^2\\times\\mathbb{R}$ is contained in the horizontal plane $t=0$ with endpoints $\\widetilde{v}_2(0)\\in\\widetilde h_3$ and $\\widetilde{v}_2(b)\\in\\widetilde h_1$. Expressing in coordinates $\\widetilde v_2(t)=(x(t),y(t),0)$, we can also assume that $(x(0),y(0))=(0,1)$ and $x(t)<0$ when $t$ is close to $0$ (since $\\widetilde v_2$ and $\\widetilde h_3$ are orthogonal), see Figure~\\ref{fig:horocycle-foliation}. We will write $(x(b),y(b))=(x_0,y_0)$ for simplicity. Recall that this parametrization comes (via conjugation) from the chosen orientation in which $v_2'=\\partial_t$.\n\nLet $\\theta\\in\\mathcal C^\\infty[0,b]$ be the angle of rotation of $\\widetilde v_2$ with respect to the horocycle foliation in the sense of~\\eqref{eqn:foliation-angle}, where we choose the initial angle $\\theta(0)=\\pi$. We will call $\\theta_0=\\theta(b)$ and assume in what follows that $\\pi<\\theta(t)<2\\pi$ and $x(t)<0$ if $0<t\\leq b$ (these inequalities will hold true by Lemma~\\ref{lem:second-period}). The complete geodesic $\\gamma\\subset\\mathbb{H}^2$ containing the projection of $\\widetilde h_1$ can be parametrized as\n\\begin{equation}\\label{eqn:gamma}\n\\gamma:(0,\\pi)\\to \\mathbb{H}^2,\\quad \\gamma(t)=\\left(x_0-y_0\\dfrac{\\cos(t)+\\cos(\\theta_0)}{\\sin(\\theta_0)}, -y_0 \\dfrac{\\sin(t)}{\\sin(\\theta_0)}\\right).\n\\end{equation}\nNote that $\\gamma(\\theta_0-\\pi)=(x_0,y_0)$ and $\\gamma'(\\theta_0-\\pi)=\\frac{-1}{\\sin(\\theta_0)}(\\sin(\\theta_0)E_1-\\cos(\\theta_0)E_2)$. If $\\gamma$ meets the $y$-axis at the point $\\gamma(t_*)$, we define the second period as the cosine of the (non-oriented) angle $\\delta$ at $\\gamma(t_*)$ subtended by the arc $\\widetilde v_2$ (see Figure~\\ref{fig:horocycle-foliation}). From the parameterization~\\eqref{eqn:gamma}, we can compute\n\\begin{equation}\\label{eqn:p2}\n\\mathcal P_2(a,\\varphi,b)=\\cos(\\delta)=\\frac{\\langle \\gamma'(t_*),E_2\\rangle}{|\\gamma'(t_*)|}=\\cos(t_*)=\\frac{x_0\\sin(\\theta_0)}{y_0}-\\cos(\\theta_0).\n\\end{equation}\nHowever, the right-hand side of~\\eqref{eqn:p2} makes sense (and we will take it as the definition of $\\mathcal P_2$) even though there is no such intersection point. Lemma~\\ref{lem:second-period} will show that if $\\mathcal P_1(a,\\varphi,b)=0$ and $\\mathcal P_2(a,\\varphi,b)=\\cos(\\frac{\\pi}{k})$ for some $k\\geq 3$, then $\\gamma$ and the positive $y$-axis do intersect with angle $\\delta=\\frac{\\pi}{k}$. \n\\end{enumerate}\n\n\\begin{figure}\n\t\\includegraphics[height=5.8cm]{fig-horocycles.pdf}\n\t\\caption{The angle $\\theta_0$ of rotation of $\\widetilde v_2$ with respect to the horocycle foliation at $\\widetilde v_2(b)$, where we identify $\\mathbb{H}^2\\times\\{0\\}$ and $\\mathbb H^2$. The surface $\\widetilde\\Sigma(a,\\varphi,b)$ projects onto the shaded region $\\widetilde\\Delta$, with boundary the projections of the labeled curves. The complete geodesic $\\gamma$ containing the projection of $\\widetilde h_1$ appears in dotted line.}\\label{fig:horocycle-foliation}\n\\end{figure}\n\nThe uniqueness of solution of the Jenkins--Serrin problem implies that $\\widetilde\\Sigma(a,\\varphi,b)$ depends smoothly on the parameters $(a,\\varphi,b)$ in the sense that, given a sequence $(a_n,\\varphi_n,b_n)$ converging to some $(a,\\varphi,b)$, the sequence of surfaces with boundary $\\widetilde\\Sigma(a_n,\\varphi_n,b_n)$ converges in the $\\mathcal{C}^k$-topology to $\\widetilde\\Sigma(a,\\varphi,b)$ for all $k\\geq 0$. This follows from standard convergence arguments for minimal graphs along with the continuity of the conjugation, see~\\cite[Proposition~2.10]{MorRod} or~\\cite[Proposition~2.3]{CMR}.\n\n\n\\subsection{Solving the period problems} In the sequel we will assume that $b$ is any non-negative real number and $(a,\\varphi)$ lies in the domain\n\\[\\Omega=\\left\\{(a,\\varphi)\\in\\mathbb{R}^2:0<\\varphi<\\tfrac{\\pi}{2},0<a<a_{\\mathrm{max}}(\\varphi)\\right\\}.\\]\nwhere\n\\begin{equation}\\label{eqn:amax}\na_{\\rm max}(\\varphi)=2\\arctanh\\left(\\tanh(l)\\cos(\\varphi)\\right),\n\\end{equation}\nand $\\tanh(l)=1$ whenever $l=\\infty$. The condition $0<a<a_{\\rm max}(\\varphi)$ means that the angle of $\\Delta$ at $p_3$ is always greater than $\\varphi$, and then an isosceles triangle $\\Delta_{0}$ with vertexes $p_2$, $p_3$, $p_4$, and $d(p_2,p_4)=d(p_3,p_4)=l$ intersects $\\Delta$ as in Figure~\\ref{propl1} (the vertex $p_4$ is ideal provided that $l=\\infty$). Note that $a_{\\rm max}(\\varphi)$ is the length of the unequal side of an isosceles triangle whose equal sides have length $l$ and whose equal angles are equal to $\\varphi$, so formula~\\eqref{eqn:amax} easily follows from the fact that the cosine of an angle of a hyperbolic right triangle is the quotient of the hyperbolic tangents of the adjacent side and the hypotenuse.\n\n\\begin{remark}\nThe restriction $(a,\\varphi)\\in\\Omega$ is used in Lemma~\\ref{lem:first-period} to compare $\\Sigma(a,\\varphi,b)$ with a solution of a Jenkins--Serrin problem over $\\Delta_0$ and solve the first period problem. Similar arguments to those in Lemma~\\ref{lem:first-period} show that if $a>a_{\\mathrm{max}}(\\varphi)$, then the first period problem has no solution, so the condition $(a,\\varphi)\\in\\Omega$ is natural.\n\\end{remark}\n\n\\begin{figure}\n\t\t\\includegraphics[height=4cm]{fig-comparison.pdf}\n\t\t\\caption{On the left, boundary values for Jenkins--Serrin problems in $\\mathbb{H}^2$ solved by $\\Sigma(a,\\varphi,b)$ and $\\Sigma_{0}(b)$, where the perpendicular bisector of $\\ell_1$ is represented in dotted line and $l<\\infty$. On the right, the limit $\\Sigma_\\infty\\subset\\mathbb{R}^3$ by rescaling (fixing the length of $\\ell_1$ equal to $1$) and the helicoid $\\Sigma_0\\subset\\mathbb{R}^3$ in the proof of Lemma~\\ref{lem:first-period}.}\\label{propl1}\n\t\\end{figure}\n\n\n\\begin{lemma}\\label{lem:p1-monotonicity}\n\t$\\mathcal P_1:\\Omega\\times\\mathbb{R}^+\\to\\mathbb{R}$ is a continuous and strictly decreasing function with respect to the third argument $b$.\n\\end{lemma}\n\n\\begin{proof}\nConsider two surfaces $\\Sigma_1=\\Sigma(a,\\varphi,b_1)$ and $\\Sigma_2=\\Sigma(a,\\varphi,b_2)$ with $0<b_1<b_2$. Let us translate each $\\Sigma_i$ vertically so that it takes the values $0$ along $\\ell_1$ and $-b_i$ along $\\ell_3$. The surface $\\Sigma_1$ lies above $\\Sigma_2$ and we can compare the vertical components of their inward-pointing conormals $\\eta_1$ and $\\eta_2$, which satisfy $\\langle \\eta_2 , \\partial_t\\rangle<\\langle \\eta_1, \\partial_t\\rangle$ in the interior of $h_1$ by the boundary maximum principle. In particular, \n\\[\\mathcal P_1(a,\\varphi,b_2)=\\int_{h_1}\\langle \\eta_2 , \\partial_t\\rangle<\\int_{h_1}\\langle \\eta_1 , \\partial_t\\rangle=\\mathcal P_1(a,\\varphi,b_1).\\qedhere\\]\n\\end{proof}\n\nIn the proof of Lemma~\\ref{lem:first-period} we need that a Jenkins--Serrin solution over $\\Delta_0$ has radial limits as we approach a vertex of $\\Delta_0$. To this end, we will state a more general result (we also point out that it extends easily to other $\\mathbb{E}(\\kappa,\\tau)$-spaces).\n\n\\begin{lemma}\\label{lem:radial-limits}\nLet $\\Omega\\subset\\mathbb{H}^2$ be an open domain with piecewise regular boundary, and assume that $\\beta_1$ and $\\beta_2$ are two regular components of $\\partial\\Omega$ that meet in a common vertex $p\\in\\partial\\Omega$ with interior angle $0<\\alpha<2\\pi$. Suppose that $u\\in\\mathcal C^\\infty(\\Omega)$ is a solution to the minimal surface equation over $\\Omega$ with bounded continuous limit values along $\\beta_1$ and asymptotic value $+\\infty$ or $-\\infty$ along $\\beta_2$. Then, $u$ has finite radial limits at $p$ along any geodesic segment interior to $\\Omega$ and not tangent to $\\beta_1$ or $\\beta_2$.\n\\end{lemma}\n\n\\begin{proof}\nLet $\\gamma$ be the vertical geodesic segment projecting onto $p$ and lying in the boundary of $\\Sigma$, the minimal surface spanned by $u$. Observe that $\\Sigma$ extends analytically across $\\gamma$ by axial symmetry, so the normal $N$ (with $\\nu>0$ at the interior of $\\Sigma$) extends smoothly to $\\gamma$. Moreover, $N$ rotates monotonically along $\\gamma$ because $\\Sigma$ is a graph, as a consequence of the boundary maximum principle for minimal surfaces. Therefore, the conormal $J\\gamma'=N\\times\\gamma'$ also rotates monotonically along $\\gamma$ (see Figure~\\ref{fig:orientation}). Since $J\\gamma'$ is horizontal and tangent to the level curves of the height function of $\\Sigma$, we deduce that the projections of such level curves form an open book foliation of a neighborhood of $p$ with binding at $p$. \n\nThis implies that, when we approach $p$ along an interior geodesic $\\sigma$ not tangent to $\\beta_1$ or $\\beta_2$, the limit of $u$ will be precisely the value of $u$ at the unique level curve (in the aforesaid foliation) tangent to $\\sigma$ at $p$, so the desired limit exists and is finite.\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem:first-period}\nThere exists a unique function $f:\\Omega\\to\\mathbb{R}_+$ such that $\\mathcal P_1(a,\\varphi,f(a,\\varphi))=0$ for all $(a,\\varphi)\\in\\Omega$. Furthermore,\n\\begin{enumerate}[label=\\emph{(\\alph*)}]\n\t\\item $f$ is a continuous function;\n\t\\item given $\\varphi_0\\in(0,\\frac{\\pi}{2})$, \n\t\\[\\lim\\limits_{a\\to a_{\\mathrm{max}}(\\varphi_0)}f(a,\\varphi_0)=+\\infty,\\qquad\\lim\\limits_{(a,\\varphi)\\to (0,\\varphi_0)}f(a,\\varphi)=0.\\]\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nFix $(a,\\varphi)\\in\\Omega$. Let $p_4$ be the point in the perpendicular bisector of the segment $\\ell_1$ with $d(p_2,p_4)=d(p_3,p_4)=l$, such that the triangle $\\Delta_0$ with vertexes $p_1$, $p_2$ and $p_4$ lies on the same side of $\\ell_1$ as $\\Delta$, see Figure~\\ref{propl1}. Let $\\Sigma_0(b)$ be the unique solution to the Jenkins--Serrin problem over the triangle $\\Delta_0$ with values $b$ along the segment $\\overline{p_2p_3}$, $+\\infty$ along $\\overline{p_3p_4}$, and $-\\infty$ along $\\overline{p_2p_4}$. Since $(a,\\varphi)\\in\\Omega$, then $\\Delta\\cap\\Delta_0\\subset\\mathbb{H}^2$ is a bounded geodesic triangle with vertexes $p_2$, $p_3$ and $q\\in\\ell_3$. Lemma~\\ref{lem:radial-limits} says that $\\Sigma_0(b)$ has a finite radial limit at $p_2$ along $\\ell_3$ as a graph over $\\Delta_0$. Therefore, if $b$ is large enough, then $\\Sigma_0(b)$ is above $\\Sigma(a,\\varphi,b)$ over the boundary of $\\Delta\\cap\\Delta_0$. By the maximum principle, it is also above $\\Sigma(a,\\varphi,b)$ in the interior of $\\Delta\\cap\\Delta_0$. In particular, we get that $\\Sigma\\cap\\Sigma_0(b)=h_1\\cup v_2\\cup v_3$ when $b$ is large. \n\nThis means that can compare the vertical components of the inward-pointing conormals $\\eta$ and $\\eta_0$ of $\\Sigma(a,\\varphi,b)$ and $\\Sigma_0(b)$, respectively, along the curve $h_1$ (as in Lemma~\\ref{lem:p1-monotonicity}). By the boundary maximum principle for minimal surfaces, we get the strict inequality $\\langle \\eta , \\partial_t\\rangle < \\langle \\eta_0 , \\partial_t\\rangle$ in the interior of $h_1$, and hence  \n\\begin{equation}\\label{prop:first-period-saddle-tower:eqn1}\n\t\\mathcal P_1(a,\\varphi,b)=\\int_{h_1}\\langle \\eta , \\partial_t\\rangle<\\int_{h_1}\\langle \\eta_0 , \\partial_t\\rangle=0,\n\\end{equation}\nprovided that $b$ is large enough. The last integral in~\\eqref{prop:first-period-saddle-tower:eqn1} vanishes because $\\Sigma_0(b)$ is axially symmetric with respect to the perpendicular bisector of $h_1$ in $\\mathbb{H}^2\\times\\{b\\}$.\n\t\nDue to the continuity of $\\Sigma(a,\\varphi,b)$ with respect to the parameters $(a,\\varphi,b)$, the surfaces $\\Sigma(a,\\varphi,b)$ converge to $\\Sigma(a,\\varphi,0)$ as $b\\to 0$. We have that $\\mathcal P_1(a,\\varphi,0)>0$ since $\\Sigma(a,\\varphi,0)$ lies above the horizontal surface $\\Delta\\times\\{0\\}$ by the maximum principle, and we can compare the third coordinate of their conormals along the common boundary $h_1$ by the boundary maximum principle (note that the third coordinate of the conormal of $\\Delta\\times\\{0\\}$ identically vanishes). By the continuity and monotonicity of $\\mathcal P_1$ with respect to $b$ proved in Lemma~\\ref{lem:p1-monotonicity}, there exist a unique $b_0\\in \\mathbb{R}^+$ such that $\\mathcal P_1(a,\\varphi,b_0)=0$. Hence this defines unequivocally $f(a,\\varphi)=b_0$. The continuity of $f$ is a consequence of its uniqueness. If $(a_n,\\varphi_n)$ and $(a_n',\\varphi_n')$ are two sequences in $\\Omega$ converging to some $(a_\\infty,\\varphi_\\infty)\\in\\Omega$ such that, after passing to a subsequence, $f(a_n,\\varphi_n)\\to b_\\infty$ and $f(a_n',\\varphi_n')\\to b_\\infty'$, and then $\\mathcal P_1(a_\\infty,\\varphi_\\infty,b_\\infty)=\\mathcal P_1(a_\\infty,\\varphi_\\infty,b_\\infty')=0$, whence $b_\\infty=b_\\infty'$, and item (a) is proved.\n\t\t\nAs for the first limit in item (b), assume by contradiction that there is a sequence $a_n\\to a_{\\mathrm{max}}(\\varphi_0)$ such that $f(a_n,\\varphi_0)$ converges, after passing to a subsequence, to some $b_\\infty\\in[0,+\\infty)$. The surface $\\Sigma_0(b_\\infty)$ lies below $\\Sigma(a_{\\mathrm{max}}(\\varphi_0),\\varphi,b_\\infty)$ as graphs over their common domain $\\Delta=\\Delta_0$ by maximum principle, because their boundary values are ordered likewise. Note that they have a common value $b_\\infty$ along $\\ell_1$, so their inward-pointing conormals can be compared along $h_1$ again by the boundary maximum principle. Since the $\\Sigma_0(b_\\infty)$ has zero period because of its symmetry, this contradicts the fact that $\\Sigma(a_{\\mathrm{max}}(\\varphi_0),\\varphi,b_\\infty)$ also has zero period.\n\t\t\nWe will compute the limit as $(a,\\varphi)$ approaches $(0,\\varphi_0)$ again by contradiction, so let us assume that there is a sequence $(a_n,\\varphi_n)$ tending to $(0,\\varphi_0)$ such that (after passing to a subsequence) $f(a_n,\\varphi_n)\\to b_\\infty$, with $b_\\infty\\in(0,+\\infty]$. Let us translate the surfaces $\\Sigma(a_n,\\varphi_n,f(a_n,\\varphi_n))$ vertically so that they take zero value along $\\ell_1$ and $-f(a_n,\\varphi_n)$ along $\\ell_3$. Since $a_n\\to 0$, we can blow up the surface and the metric of $\\mathbb{H}^2\\times\\mathbb{R}$ in such a way $a_n$ is equal to $1$. The new sequence of rescaled surfaces converges in the $\\mathcal C^k$-topology for all $k$ to a minimal surface $\\Sigma_\\infty$ in  Euclidean space $\\mathbb{R}^3$. This surface $\\Sigma_\\infty$ is a graph over a domain of $\\mathbb{R}^2$ bounded by three lines $\\ell_{1\\infty}$, $\\ell_{2\\infty}$ and $\\ell_{3\\infty}$ such that $\\ell_{2\\infty}$ and $\\ell_{3\\infty}$ are parallel and $\\ell_{1\\infty}$ makes an angle of $\\varphi_0$ with $\\ell_{2\\infty}$. Moreover, $\\Sigma_\\infty$ takes values $+\\infty$ along $\\ell_{2\\infty}$, $-\\infty$ along $\\ell_{3\\infty}$ (since $b_\\infty>0$), and $0$ along $\\ell_{1\\infty}$. Let us consider $\\Sigma_0$ the helicoid of $\\mathbb{R}^3$ with axis $\\ell_{1\\infty}$ which is a graph over a half-strip of $\\mathbb{R}^2$ as depicted in Figure~\\ref{propl1} (right). Since $0<\\varphi_0<\\frac{\\pi}{2}$, the intersection of the domains of $\\Sigma_0$ and $\\Sigma_\\infty$ is a triangle on whose sides the boundary values of $\\Sigma_0$ are greater than or equal to the corresponding values of $\\Sigma_\\infty$. By the maximum principle, we deduce that $\\Sigma_0$ lies above the surface $\\Sigma_\\infty$ also in the interior of that triangle. Hence, we can compare their conormals along $\\ell_{1\\infty}$ by the boundary maximum principle to conclude that the period of $\\Sigma_\\infty$ is not zero, which contradicts that each of the surfaces $\\Sigma(a_n,\\varphi_n,f(a_n,\\varphi_n))$ has zero period.\\qedhere\n\\end{proof}\n\nThis solves the first period problem, and we will now focus on the second one. To this end, we will use the notation defined in Section~\\ref{subsec:periods} (see also Figure~\\ref{fig:horocycle-foliation}).\n\n\\begin{lemma}\\label{lem:second-period}\n\tLet $\\varphi_0\\in(0,\\frac{\\pi}{2})$ and $a\\in(0,a_{\\mathrm{max}}(\\varphi_0))$.\n\t\\begin{enumerate}[label=\\emph{(\\alph*)}]\n\t\t\\item The inequalities $x(t)<0$ and $\\pi<\\theta(t)<2\\pi$ hold true for all $t\\in(0, b]$.\n\t\t\\item If the curve $\\gamma$ intersects the positive $y$-axis with angle $\\delta$, then $\\delta<\\varphi_0$, in which case $\\mathcal P_2(a,\\varphi_0,f(a,\\varphi_0))=\\cos(\\delta)$.\n\t\t\\item If $\\mathcal P_2(a,\\varphi_0,f(a,\\varphi_0))=\\cos(\\delta)$ for some $\\delta\\in(0,\\varphi_0)$, then $\\gamma$ intersects the positive $y$-axis with angle $\\delta$.\n\t\t\\item If $\\mathcal P_2(a,\\varphi_0,f(a,\\varphi_0))=1$, then $\\gamma$ and the $y$-axis are asymptotic geodesics intersecting at the ideal point $(0,0)$.\n\t\\end{enumerate}\n\tFurthermore,\n\t\\[\\lim_{a\\to 0}\\mathcal P_2(a,\\varphi_0,f(a,\\varphi_0))=\\cos(\\varphi_0),\\qquad \n\t\\lim_{a\\to a_{\\mathrm{max}}(\\varphi_0)}\\mathcal P_2(a,\\varphi_0,f(a,\\varphi_0))=+\\infty.\\]\n\\end{lemma}\n\n\n\\begin{proof}\nWe will identify $\\widetilde v_2$ with its projection to $\\mathbb H^2$ for the sake of simplicity. Therefore, $\\widetilde v_2$ is strictly convex (in the hyperbolic geometry) towards the exterior of $\\widetilde\\Delta$ by Lemma~\\ref{lem:orientation}, and this implies that any geodesic tangent to $\\widetilde v_2$ lies locally in the interior of $\\widetilde\\Delta$ except for the point of tangency. In particular, we have that $\\theta(t)>\\pi$ for $t$ close to $0$ by just comparing $\\widetilde v_2$ with the tangent geodesic at $\\widetilde v_2(0)=(0,1)$ (see Figure~\\ref{fig:gauss-bonnet}, left). Furthermore, if $\\theta(t)>\\pi$ does not hold for all $t\\in(0,b]$, then at the smallest $t_0>0$ such that $\\theta(t_0)=\\pi$, the tangent geodesic has points outside $\\widetilde\\Delta$ arbitrarily close to $\\widetilde v_2(t_0)$, which is a contradiction (see Figure~\\ref{fig:gauss-bonnet}, left).\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=\\textwidth]{fig-gauss-bonnet.pdf}\n\\caption{Tangent geodesics at $\\widetilde v_2(0)$ and at a first $t_0\\in(0,b]$ such that $\\theta(t_0)=\\pi$ (left). A first $t_0\\in(0,b]$ such that $x(t_0)=0$ (center). A first $t_0\\in(0,b]$ such that $\\theta(t_0)=2\\pi$ (right). The domains $U$ and $V$ are those where we apply Gauss--Bonnet formula in Lemma~\\ref{lem:second-period}.}\\label{fig:gauss-bonnet}\n\\end{center}\n\\end{figure}\n\nAssume by contradiction that $x(t)<0$ does not hold in general, and let $t_0\\in(0,b]$ be the smallest value such that $x(t_0)=0$, so the curve $\\widetilde v_2$ between $0$ and $t_0$ together with a segment of the $y$-axis enclose a bounded domain $U\\subset\\mathbb{H}^2$ (see Figure~\\ref{fig:gauss-bonnet}, center). The curve $\\widetilde v_2$ is convex towards the interior of $U$, and $U$ has two interior angles equal to $\\frac\\pi2$ and $\\alpha=\\theta(t_0)-\\frac{3\\pi}{2}\\in(0,\\pi)$, so Gauss--Bonnet formula yields \n\\begin{equation}\\label{lem:second-period:eqn1}\n\\begin{aligned}\n0>-\\area(U)&=2\\pi+\\int_{0}^{t_0}\\kappa_g(t)\\,\\mathrm{d}t-(\\tfrac{\\pi}{2}+\\pi-\\alpha)\\\\\n&>\\tfrac{\\pi}{2}+\\int_{0}^{b}\\kappa_g(t)\\,\\mathrm{d}t=\\tfrac{\\pi}{2}-\\int_{0}^{b}\\psi'(t)\\,\\mathrm{d}t=\\tfrac{\\pi}{2}-\\varphi_0,\n\\end{aligned}\\end{equation}\nwhere $\\kappa_g<0$ is the geodesic curvature with respect to the unit conormal $\\widetilde N$ pointing outside $\\widetilde\\Delta$, see Lemma~\\ref{lem:orientation}. We have also used that the angle $\\psi$ of the normal $N$ along $v_2$, in the sense of~\\eqref{eqn:rotation-angle}, rotates counterclockwise with $\\psi'=-\\kappa_g>0$ and $\\psi(b)-\\psi(0)=\\varphi_0$. The inequality~\\eqref{lem:second-period:eqn1} contradicts the assumption $\\varphi_0\\in(0,\\frac\\pi2)$.\n\nLet us assume, again by contradiction, that there is (a first) $t_0\\in(0,b]$ such that $\\theta(t_0)=2\\pi$. This implies that the normal geodesic to $\\widetilde v_2$ at $t_0$ is a straight line parallel to the $y$-axis. Let $V\\subset\\mathbb{H}^2$ be the domain enclosed by this line together with an arc of $\\widetilde v_2$ (see Figure~\\ref{fig:gauss-bonnet}, right). Note that $V$ has two interior angles $\\alpha\\in(0,\\pi)$ and $\\frac{\\pi}{2}$, plus $\\widetilde v_2$ is convex towards $V$. Reasoning as in~\\eqref{lem:second-period:eqn1}, we get the same contradiction $0>-\\area(V)>\\frac{\\pi}{2}-\\varphi_0$, which finishes the proof of item (a).\n\nAs for item (b), if $\\gamma$ intersects the $y$-axis with angle $\\delta$, then there is a region $W\\subset\\mathbb{H}^2$ bounded by $\\gamma$, $\\widetilde v_2$ and the $y$-axis. Gauss--Bonnet formula, in the same fashion as in Equation~\\eqref{lem:second-period:eqn1}, gives the inequality $0>-\\area(W)>\\delta-\\varphi_0$, which is equivalent to $\\delta<\\varphi_0$. The equality $\\mathcal P_2(a,\\varphi,f(a,\\varphi))=\\cos(\\delta)$ was given in~\\eqref{eqn:p2}.\n\nWe will now discuss items (c) and (d). Note that $\\gamma(\\pi)$ has negative first coordinate by the above analysis, so $\\gamma$ intersects the $y$-axis if and only the first coordinate of\n\\begin{equation}\\label{prop:second-period:eqn1}\n\\gamma(0)=\\left(x_0-y_0\\frac{1+\\cos(\\theta_0)}{\\sin(\\theta_0)},0\\right)\n\\end{equation}\nis positive (here, $\\sin(\\theta_0)<0$ because $\\pi<\\theta_0<2\\pi$). If there exists $\\delta\\in(0,\\varphi_0)$ such that $\\mathcal P_2(a,\\varphi_0,f(a,\\varphi_0))=\\frac{x_0\\sin(\\theta_0)}{y_0}-\\cos(\\theta_0)=\\cos(\\delta)\\in (0,1)$, then the first coordinate in~\\eqref{prop:second-period:eqn1} is positive, and it follows from (b) that the angle at the intersection is precisely $\\delta$. If $\\mathcal P_2(a,\\varphi_0,f(a,\\varphi_0))=1$, then the first coordinate of~\\eqref{prop:second-period:eqn1} vanishes, so $\\gamma$ and the $y$-axis are asymptotic at the ideal point $(0,0)$.\n\nTo finish the proof, let us analyze the limits. Integrating from $0$ to $b$ the identity $\\theta'=\\psi'-\\cos(\\theta)$ in Lemma~\\ref{lem:orientation} (applied to $v_2$), and taking into account that $\\theta(b)-\\theta(0)=\\theta_0-\\pi$ and $\\psi(b)-\\psi(0)=\\varphi_0$, we get the relation\n\\begin{equation}\\label{eqn:twist2}\n\\theta_0=\\varphi_0+\\pi-\\int_0^b\\cos(\\theta(s))\\,\\mathrm{d}s.\n\\end{equation}\nIn particular, $\\theta_0\\to\\varphi_0+\\pi$ and $(x_0,y_0)\\to(0,1)$ as $b\\to 0$ (note that the length of $\\widetilde v_2$ goes to zero). This implies that the first component of~\\eqref{prop:second-period:eqn1} is positive, i.e., $\\gamma(0)$ and $\\gamma(\\pi)$ lie at distinct sides of the $y$-axis for $b$ small enough, so $\\gamma$ intersects the positive $y$-axis at some point. By Lemma~\\ref{lem:first-period}, if $a\\in(0,a_{\\mathrm{max}}(\\varphi_0))$ tends to zero, then $b=f(a,\\varphi_0)$ also tends to zero and \n\\[\\lim_{a\\to 0}\\mathcal P_2(a,\\varphi_0,f(a,\\varphi_0))=\\lim_{a\\to 0}\\left(\\frac{x_0\\sin(\\theta_0)}{y_0}-\\cos(\\theta_0)\\right)=\\cos(\\varphi_0).\\]\n\t\nAs for the limit $a\\to a_{\\mathrm{max}}(\\varphi_0)$, let $(a_n,\\varphi_0)\\in\\Omega$ be a sequence with $a_n\\to a_{\\mathrm{max}}(\\varphi_0)$. Lemma~\\ref{lem:first-period} tells us that $b_n=f(a_n,\\varphi_0)\\to +\\infty$, so the surfaces $\\Sigma(a_n,\\varphi_0,b_n)$ converge, up to a subsequence and vertical translations (in such a way $h_1$ is a segment at height $0$) to a solution $\\Sigma_\\infty$ of a Jenkins--Serrin problem over an isosceles triangle with values $0$ along the unequal side and $+\\infty$ and $-\\infty$ along the other sides. We will denote in the sequel the elements of $\\Sigma(a_n,\\varphi_0,b_n)$ with a subindex $n$.\n\\begin{itemize}\n\t\\item If $l<\\infty$, the conjugate surfaces converge to $\\widetilde\\Sigma_\\infty$, twice the fundamental piece of a symmetric saddle tower with four ends in the quotient (this conjugate construction is analyzed in~\\cite{MorRod}). If we fix $\\widetilde v_{2n}\\subset\\mathbb{H}^2\\times\\{0\\}$, then the curves $\\widetilde{v}_{1n}$ converge to a complete horizontal curve $\\widetilde v_{1\\infty}\\subset\\mathbb{H}^2\\times\\{-l\\}$ (convex towards the exterior of the domain), and the curves $\\widetilde h_{3n}$ tend to an ideal vertical segment $\\widetilde h_{3\\infty}$, see Figure~\\ref{fig:catenoid}.\tHowever, we will translate and rotate the surfaces first so that $\\widetilde v_{2n}(0)=(0,1,0)$ and $\\widetilde v_{2n}'(0)=-\\partial_x$ in the half-space model in order to analyze the rotation $\\theta_{0n}$ of $\\widetilde v_{2n}'$ with respect to the horocycle foliation (i.e., we adapt the sequence to the setting of Figure~\\ref{fig:horocycle-foliation}). This means that a subsequence of $\\Sigma(a_n,\\varphi_0,b_n)$ no longer converges to a saddle tower but to a subset of the vertical plane $x^2+y^2=1$. Therefore, $\\theta_{0n}\\to\\frac{3\\pi}{2}$ and $\\widetilde v_{2n}(b_n)=(x_{0n},y_{0n})\\to(-1,0)$ as $n\\to\\infty$. In view of~\\eqref{prop:second-period:eqn1}, we deduce that $\\gamma_n$ does not intersect the positive $y$-axis for large $n$, and~\\eqref{eqn:p2} implies that $\\mathcal P_2(a_n,\\varphi_0,b_n)\\to+\\infty$.\n\n\t\\item If $l=\\infty$, then it is also well known~\\cite{MorRod,Pyo} that the conjugate surfaces converge to $\\widetilde\\Sigma_\\infty$, a quarter of a horizontal catenoid when we keep the point $\\widetilde v_{2n}(b_n)$ fixed (and hence  the curves $\\widetilde{v}_{1n}$ converge to a complete ideal horizontal geodesic $\\widetilde v_{1\\infty}\\subset\\mathbb{H}^2\\times\\{-\\infty\\}$). However, if we fix $\\widetilde v_{2n}(0)=(0,1,0)$ and $\\widetilde v_{2n}'(0)=-\\partial_x$ instead, then a subsequence converges to a subset of the vertical plane $x^2+y^2=1$ as in the case $l<\\infty$, so we can reason likewise.\\qedhere\n\\end{itemize}\n\\end{proof}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{fig-catenoid.pdf}\n\\caption{The limit saddle tower ($l<\\infty$) and catenoid ($l=\\infty$) when $a\\to a_{\\rm max}(\\varphi_0)$. In the proof Lemma~\\ref{lem:second-period} we bring the points at which the arrows aim to a fixed point of $\\mathbb H^2$, so we get vertical planes in the limit (instead of the saddle tower or the catenoid).}\\label{fig:catenoid}\n\\end{center}\n\\end{figure}\n\n\n\\begin{proof}[Proof of Theorems~\\ref{thm:knoids} and~\\ref{thm:saddle-towers}]\nLet $k\\geq 3$. For each $\\frac{\\pi}{k}<\\varphi<\\frac{\\pi}{2}$, Lemma~\\ref{lem:second-period} ensures that $\\mathcal P_2(a,\\varphi,f(a,\\varphi))$ tends to $\\cos(\\varphi)$ when $a\\to 0$ and tends to $+\\infty$ when $a\\to a_{\\mathrm{max}}(\\varphi)$. Since $\\cos(\\varphi)<\\cos(\\frac{\\pi}{k})$ and $\\mathcal P_2$ is continuous, there exists some $a_\\varphi\\in(0,a_{\\mathrm{max}}(\\varphi))$ such that $\\mathcal P_2(a,\\varphi,f(a_\\varphi,\\varphi))=\\cos(\\frac{\\pi}{k})$, though it might not be unique. Therefore, we deduce from item (c) of Lemma~\\ref{lem:second-period} that $\\widetilde\\Sigma_\\varphi=\\widetilde\\Sigma(a_\\varphi,\\varphi,f(a_\\varphi,\\varphi))$ solves both period problems. We will show that $\\Sigma_\\varphi=\\Sigma(a_\\varphi,\\varphi,f(a,\\varphi))$, and hence $\\widetilde\\Sigma_\\varphi$, has finite total curvature by adapting Collin and Rosenberg's argument, see~\\cite[Remark~7]{CR}.\n\nTo this end, we consider first the case $l=\\infty$. For each $k\\in\\mathbb{N}$, let $p_1(k)\\in\\ell_3$ be such that $d(p_1(k),p_3)=k$, and let $\\ell_2(k)$ (resp.\\ $\\ell_3(k)$) be the geodesic segment joining $p_3$ and $p_1(k)$ (resp.\\ $p_2$ and $p_1(k)$). For each $n\\in\\mathbb{N}$ with $n\\geq f(a_\\varphi,\\varphi)$, we will consider the Dirichlet problem over the triangle of vertexes $p_1(k)$, $p_2$ and $p_3$ with boundary values $f(a_\\varphi,\\varphi)$ on $\\ell_1$, $n$ on $\\ell_2(k)$ and $0$ over $\\ell_3(k)$. These conditions span a unique compact minimal disk $\\Sigma_\\varphi^{k,n}$ (with boundary) which is a graph over the interior of the triangle. The surface $\\Sigma_\\varphi^{k,n}$ has geodesic boundary and six internal angles, all of them equal to $\\frac{\\pi}{2}$, so Gauss--Bonnet formula gives a total curvature of $-\\pi$ for $\\Sigma_\\varphi^{k,n}$. As $k\\to\\infty$, the surfaces $\\Sigma_\\varphi^{k,n}$ converge uniformly on compact subsets (as graphs) to a surface $\\Sigma_\\varphi^n$ over $\\Delta$ with boundary values $f(a_\\varphi,\\varphi)$ over $\\ell_1$, $n$ over $\\ell_2$, and $0$ over $\\ell_3$ (this convergence is monotonic by the maximum principle). Therefore, Fatou's Lemma implies that the total curvature of $\\Sigma_\\varphi^{n}$ is at least $-\\pi$. Finally, we let $n\\to\\infty$ so the $\\Sigma_\\varphi^{n}$ converge (also monotonically) to $\\Sigma_\\varphi$ on compact subsets, and the same argument implies that $\\Sigma_\\varphi$ has finite total curvature at least $-\\pi$ (note that the Gauss curvature of a minimal surface in $\\mathbb{H}^2\\times\\mathbb{R}$ is nowhere positive by Gauss equation). If $l<\\infty$, the same idea works by just truncating at height $n$ (i.e., there is no need of introducing the sequence with index $k$).\n\nBy successive mirror symmetries across the planes containing the components of $\\partial\\widetilde\\Sigma_\\varphi$, we get a complete proper Alexandrov-embedded minimal surface $\\overline\\Sigma_\\varphi\\subset\\mathbb{H}^2\\times\\mathbb{R}$.\n\n\n\\begin{itemize}\n\t\\item  If $l=\\infty$, then the curve $\\widetilde v_1$ is an ideal horizontal geodesic, and we only need to reflect once about a horizontal plane, i.e., the plane containing $\\widetilde v_2$ and $\\widetilde v_3$. Hence, $\\overline\\Sigma_\\varphi$ consists of $4k$ copies of $\\widetilde\\Sigma_\\varphi$, so the total curvature in this case is not less than $-4k\\pi$, and~\\cite[Theorem~4]{HMR} ensures that $\\overline\\Sigma_\\varphi$ is asymptotic to a certain geodesic polygon at infinity. From the above analysis, each end of $\\overline\\Sigma_\\varphi$ has asymptotic boundary consisting of four complete ideal geodesics: two horizontal ones obtained from $\\widetilde v_1$, and two vertical ones obtained from $\\widetilde h_2$. Taking into account that $\\overline\\Sigma_\\varphi$ has genus $g=1$, Equation~\\eqref{eqn:generalized-GB} (with $m=k$) reveals that its total curvature is exactly $-4k\\pi$. \n\n\tNote that each end of $\\overline\\Sigma_\\varphi$ is asymptotic to a vertical plane and it is contained in four copies of $\\widetilde\\Sigma_\\varphi$. We claim that the subset of $\\overline\\Sigma_\\varphi$ formed by these four copies is a symmetric bigraph, so the end is embedded in particular. This claim follows from the fact that two of these four pieces come from $\\Sigma_\\varphi$ and its axially symmetric surface with respect to $h_2$, which project to a quadrilateral of $\\mathbb{H}^2$. Since this quadrilateral is convex, the Krust-type result in~\\cite{HST} guarantees that the conjugate $\\widetilde\\Sigma_\\varphi$ and its mirror symmetric surface across $\\widetilde h_3$ form a graph. The other two copies needed to produce the aforesaid bigraph are their symmetric ones with respect to the slice containing $\\widetilde v_2$ and $\\widetilde v_3$.\n\n\t\\item If $l<\\infty$, then the composition of the reflections with respect to the horizontal planes containing $\\widetilde v_1$ and $\\widetilde v_3$ is a vertical translation $T$ of length $2l$. Thus, $\\overline\\Sigma_\\varphi$ induces a surface in the quotient of $\\mathbb{H}^2\\times\\mathbb{R}$ by $T$ with total Gauss curvature at least $-4k\\pi$, since it consists of $4k$ pieces isometric to $\\widetilde\\Sigma_\\varphi$. This surface has genus $1$ and $2k$ ends, so it follows from the main theorem in~\\cite{HM} that its total curvature is exactly $-4k\\pi$. This result also implies that each end of $\\overline\\Sigma_\\varphi$ is asymptotic to a vertical plane (in the quotient). \\qedhere\n\\end{itemize}\n\\end{proof}\n\n \\begin{remark}\n It is important to notice that we have not proved the uniqueness of the surface $\\Sigma_\\varphi$. This would be automatically true if we could show that the second period $\\mathcal P_2(a,\\varphi,f(a,\\varphi))$ is strictly increasing in the parameter $a$, though a comparison of the surfaces for different values of $a$ seems to be difficult, since we do not even know if the function $f$ solving the first period problem is monotonic.\n \\end{remark}\n\nAs $\\varphi$ approaches $\\frac{\\pi}{k}$, the value $a_\\varphi$ solving the two period problems in the proof of Theorems~\\ref{thm:knoids} and~\\ref{thm:saddle-towers} goes to zero, and the surface $\\widetilde\\Sigma_\\varphi$ converges, after rescaling, to a genus $1$ minimal $k$-noid in $\\mathbb{R}^3$ (as in item (b) of Lemma~\\ref{lem:first-period}). Moreover, when $\\varphi$ approaches $\\frac{\\pi}{2}$, the surface $\\Sigma_\\varphi$ converges to an open subset of a helicoid in $\\mathbb{R}^3$ after rescaling, and it follows that the conjugate surfaces $\\widetilde\\Sigma_\\varphi$ must converge a quarter of a catenoid in $\\mathbb{R}^3$ (the curve $\\widetilde h_1$ converges to half of the neck of such catenoid).\n\n\\subsection{The embeddedness problem}\n\nIn the proof of Theorems~\\ref{thm:knoids} and~\\ref{thm:saddle-towers}, it is shown that the conjugate piece $\\widetilde{\\Sigma}_\\varphi$ is a graph over the domain $\\widetilde \\Delta\\subset\\mathbb{H}^2$. But it could happen that when we reflect $\\widetilde{\\Sigma}_\\varphi$ over the vertical plane containing $\\widetilde h_1$, the resulting surface is not embedded since the reflected curve of $\\widetilde v_3$ might intersect $\\widetilde v_3$. Observe that, as the family of examples with $k$ ends converges to a genus $1$ minimal $k$-noid in $\\mathbb{R}^3$ after blow up, see also~\\cite{Maz}, there do exist non-embedded examples of $k$-noids and saddle towers with genus $1$ in $\\mathbb{H}^2\\times\\mathbb{R}$ for all $k\\geq 3$.\n\n\\begin{figure}\n\t\\includegraphics[width=\\textwidth]{fig-embedding.pdf}\n\t\\caption{Graphics of the functions $\\varphi\\mapsto a_{\\text{max}}(\\varphi)$ and $\\varphi\\mapsto a_{\\text{emb}}(\\varphi)$ with $l=\\infty$ (left) and $l=1$ (right). In the shaded regions, embeddedness is guaranteed by the Krust property.}\\label{aemb} \n\\end{figure}\n\nTherefore, embeddedness is guaranteed if the extended surface by reflection about the vertical plane containing $\\widetilde h_1$ is embedded. The Krust property yields this if the initial surface $\\Sigma_\\varphi$ extended by axial symmetry about the geodesic $h_1$ is still a graph over a convex domain, i.e., if the angle of $\\Delta$ at $p_3$ is at most $\\frac{\\pi}{2}$. Elementary hyperbolic geometry shows that this is equivalent to $a\\geq a_{\\text{emb}}(\\varphi)$, where\n\t\\begin{equation}\\label{eqn:emb}\n\t\t a_{\\text{emb}}(\\varphi)=\\arcsinh(\\tanh(l)\\cot(\\varphi)),\n\t\\end{equation}\nand $\\tanh(l)=1$ if $l=\\infty$. Hence, the surfaces in Theorems~\\ref{thm:knoids} and~\\ref{thm:saddle-towers} are properly embedded provided that $a_{\\mathrm{emb}}(\\varphi)\\leq a_\\varphi<a_{\\text{max}}(\\varphi)$. If $l=\\infty$, then $a_{\\text{emb}}(\\varphi)<a_{\\text{max}}(\\varphi)$ for all $\\varphi\\in(0,\\frac\\pi2)$; if $l<\\infty$, there exists $\\varphi_1\\in(0,\\frac\\pi2)$  such that $a_{\\text{emb}}(\\varphi)<a_{\\text{max}}(\\varphi)$ if and only if $\\varphi\\in(\\varphi_1,\\frac\\pi2)$, see Figure~\\ref{aemb}.\n\nHowever, on the one hand, it seems difficult to know if a value of $a_\\varphi$ solving both period problems lies in this interval; on the other hand, embeddedness may occur even if $a\\geq a_{\\rm emb}(\\varphi)$ does not hold. It is expected that there are always values of $(a,\\varphi)$ producing embedded examples solving the two period problems for all $k\\geq 3$, and it seems reasonable that this occurs when $\\varphi$ becomes close to $\\frac{\\pi}{2}$.\n\n\\subsection{Examples with infinitely many ends.}\nLet us tackle the proof of Theorem~\\ref{thm:infty-noids}, which is a particular case of the above constructions for $l=\\infty$ (the proof can be easily adapted to the case $l<\\infty$). Lemma~\\ref{lem:second-period} and the continuity of $\\mathcal P_2$ imply that, for all $\\varphi\\in(0,\\frac\\pi2)$, there are values of $a\\in(0,a_{\\mathrm{max}}(\\varphi))$ such that $\\mathcal P_2(a,\\varphi,f(a,\\varphi))$ is either equal to $1$ or greater than $1$. Let us study these two cases:\n\\begin{itemize}\n\t\\item If $\\mathcal P_2(a,\\varphi,f(a,\\varphi))=1$, then $\\widetilde h_1$ and $\\widetilde h_3$ are contained in vertical planes over asymptotic geodesics of $\\mathbb{H}^2$ in view of item (d) of Lemma~\\ref{lem:second-period}. This means that $\\widetilde\\Sigma(a,\\varphi,f(a,\\varphi))$ is contained in the region between these two geodesics, see Figure~\\ref{fig:infty-noids} (center), and mirror symmetries across the corresponding vertical planes span a group of isometries fixing the common point at infinity. This group contains a discrete group of parabolic translations, and gives rise to the $1$-parameter family of parabolic $\\infty$-noids.\n\t\\item The case $\\mathcal P_2(a,\\varphi,f(a,\\varphi))>1$ occurs in an open subset of $\\Omega$, and gives rise to the $2$-parameter family of hyperbolic $\\infty$-noids. The two geodesics of $\\mathbb{H}^2$ containing the projections of $\\widetilde h_1$ and $\\widetilde h_3$ do not intersect in this case and successive reflections across their associated vertical planes span a group of isometries containing a discrete group of hyperbolic translations, see Figure~\\ref{fig:infty-noids} (right).\n\\end{itemize}\n\n\\begin{figure}[t]\n\t\\includegraphics[width=\\textwidth]{fig-infty-noids.pdf}\n\t\\caption{The fundamental domains of a $3$-noid (left), a parabolic $\\infty$-noid (middle), and a hyperbolic $\\infty$-noid (right). Dotted curves represent geodesics containing the projection of $\\widetilde h_1$ and $\\widetilde h_3$.} \\label{fig:infty-noids}\n\\end{figure}\n\nSimilar arguments to those in the proof of Theorems~\\ref{thm:knoids} and~\\ref{thm:saddle-towers} (using the description of periodic surfaces with finite total curvature in~\\cite{HM}) show that each end of the constructed surfaces is embedded and has finite total curvature, plus the global surface is Alexandrov-embedded. Observe that in the case of hyperbolic $\\infty$-noids, we can always choose $a\\geq a_{\\mathrm{emb}}(\\varphi)$, defined in the previous section, which means that whenever the parameters $(a,\\varphi)$ lie in this open subset of $\\Omega$, the reflected surface is a properly embedded hyperbolic $\\infty$-noid. In the case of parabolic $\\infty$-noids, we are not able to guarantee global embeddedness.\n\n\\medskip\n\n\\noindent\\textbf{Competing interests:} The authors declare none.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nIt is well-known that the distribution of the  \natomic hydrogen gas (HI) in many spiral galaxies is non-axisymmetric or lopsided. This lopsidedness was first studied by Baldwin, Lynden-Bell, \\& Sancisi (1980).\nFrom the analysis of global profiles of HI using single dish radio \ntelescopes, it was concluded that over 50 \\% of galaxies studied show \nsuch asymmetry (Richter \\& Sancisi 1994, Haynes et al. 1998, \nMatthews, van Driel, \\& Gallagher 1998). Due to the lack of\nspatial resolution, such studies could only reveal the combined effect of spatial and velocity asymmetry.\nThe lopsidedness has also been observed kinematically, as in terms of \nasymmetric rotation curves (Swaters et al. 1999) which can be\ndirectly used to obtain the lopsided perturbation potential\n(Jog 2002). The kinematical asymmetry has been used to deduce the\n ellipticity of the potential\nfrom the analysis of the HI velocity fields (Franx, van Gorkom, \\& de Zeeuw \n1994; Schoenmakers et al. 1997). \n\nMany theories have been proposed for the observed lopsidedness. They include \ntidal interactions (Jog 1997), minor mergers \\citep{Zaritsky97}, asymmetric gas accretion \\citep{Bournaud05}, \nand offset stellar disc in the halo potential \\citep{Noord01}. It is not yet clear which of these possible scenarios work at scales of a galactic group. \n\nRecently, in a first such study, the 2-D spatial distribution of HI was \nFourier-analysed to measure the lopsidedness for a sample of 18 galaxies in the \nEridanus group of galaxies (Angiras et al. 2006, here onwards called Paper I). \nIn this paper, it was found that the mean amplitude of measured lopsidedness \nis large $\\sim 0.2$ (i.e., a surface density contrast of 20\\% above an uniform disc), nearly twice of that seen in the stellar component \nof the field galaxies (Rix \\& Zaritsky 1995, Bournaud et al. 2005).\nAlso, it was shown that the early-type spiral galaxies \nshow a higher lopsidedness than the late-type spirals, which is contrary\nto that observed in field galaxies (Bournaud et al. 2005). These two results suggest \nthat tidal interactions play a dominant role in the generation of lopsidedness in \nthe galaxies in groups, as was argued in paper I.\nIt is not known whether these results are common to all group\nenvironments, especially since the groups are known to exhibit a\nlarge variety of properties \\citep{Rasmussen06}, especially regarding the galaxy density and velocity dispersions.\n\nHere we address this issue by analysing the 2-D HI data for \ngalaxies in the Ursa Major group. The velocity dispersion, the fraction of early type galaxies and the lack of HI deficiency \\citep{Marc01,Omar05} are the major differences between the Ursa Major and the Eridanus groups.\nThus the galaxies in the Ursa Major group provide an opportunity to study lopsidedness in a different physical environment. We have selected 11 spiral galaxies belonging to the Ursa Major Group for estimating the spatial \nlopsidedness and elongation.  The spatial lopsidedness values are compared with the results of the Eridanus group of galaxies (Paper I) to see whether group environments give rise to similar values for the amplitudes and phases of the spatial asymmetries. The higher order Fourier components (m=1,2,3) have also been obtained. Further, we have carried out an analysis of the kinematical data to estimate the elongation in the potential. This is compared with  the values obtained from the spatial analysis.As in the Eridanus case (Paper I), the use of HI as a tracer allows us to study lopsidedness to outer disks covering a radial distance larger than twice that studied earlier using the near-IR data on stars as in Rix \\& Zaritsky (1995) and Bournaud et al. (2005).\n\n\nThis paper is organised as follows. In section 2 we discuss the HI and optical data used for the proposed analysis. Details of the\nharmonic analysis, and the results are presented in section 3. A discussion of these results in the context of the group environment is given in section 4 and in section 5 we give the conclusions.\n\n\\section{Data}\n\\subsection{The Ursa Major Group}\nThe Ursa Major group in the super galactic co-ordinate system lies between $58.53\\le SGL \\le 73.53$ degrees and\n$-4.46 \\le SGB \\le 10.54$ degrees \\cite{Tully96}. The systemic velocity of this group is $950$ km s$^{-1}$ with a\ndispersion of $150 $km s$^{-1}$ \\cite{Marc01}. Seventy nine galaxies have been associated with this group \\cite{Tully96} and this group had a projected density of 3 galaxies per Mpc$^{-2}$. Due to the small velocity dispersion, it is not yet clear whether it can be classified as a cluster as mentioned in Tully et al.(1996). In addition this\ngroup contains a smaller fraction of early-type galaxies ($\\sim 14\\%$ (E+S0) and a larger fraction of late-type \ngalaxies $\\sim 86\\%$ (Sp+Irr)\\cite{Tully96}) This system does \nnot show a central concentration \\cite{Marc01} which is atypical for a cluster and more similar to that found in a group.\n\nAs we shall be using the results from a similar analysis carried out on Eridanus group by Angiras et al. (2006) \nin section 3\\&4, a brief summary of the characteristics of this group is given. This group in super-galactic \nco-ordinate system lies between $\\sim -30\\le SGL \\le -52$ degrees and $272\\ge SGB \\ge 292$ degrees at \na mean distance of $\\sim 23\\pm2$Mpc. Approximately 200 galaxies are associated with this group with a velocity \ndispersion of $\\sim 240$kms$^{-1}$. In this case the projected density was higher, $\\sim$ 8 galaxies per\n Mpc$^{-2}$ \\citep{Omar05}. \nUnlike Ursa Major, in Eridanus group there was a larger fraction of elliptical and S0 type galaxies ($\\sim 30\\%$ (E+S0) and a smaller fraction of the late-type galaxies $\\sim 70\\%$ (Sp+Irr) \\citep{Omar05}). In this case also, like Ursa Major, the group centre is not known. The selection criteria for the 18 galaxies that were used for the harmonic analysis by Angiras et al. (2006) are similar to what is given in section 2.2 (paragraph 1). \n\nEven though, these two groups are almost at the same distance from us they differ mainly in two aspects. Firstly, in Eridanus group, HI deficiency is seen which is ascribed to tidal interactions \\citep{Omar05b}. HI deficiency is not seen in the case of Ursa Major \\citep{Marc01}. Secondly, compared to Eridanus, Ursa Major is a loose group \\citep{Tully96,Omar05}.\n\nThe higher number density and the higher velocity dispersion as seen in the Eridanus group implies a higher rate of tidal interactions between the galaxies in the group. Also, the\nhigher fraction of early-type galaxies seen in the Eridanus represent an earlier evolution of galaxies via tidal \ninteractions. These agree with the higher amplitude of lopsidedness seen in the Eridanus \ngalaxies, if generated by tidal interactions.\nWe caution, however, that the spatial distribution of the Ursa Major galaxies is that of \nan elongated filament \\citep{Tully96}. Hence, it may not be so straightforward to calculate the dependence of the galaxy \ninteraction rate on the number density in that case.\n\n\n\\subsection{Radio Data}\n\nOut of the 49 galaxies observed using the Westerbork Synthesis Radio Telescope (WSRT) by Verheijen and Sancisi (2001) , we have\nselected 11 galaxies on the basis of their inclination and quality of HI maps (galaxies, whose HI maps were patchy were rejected) for further analysis. The right ascension ($\\alpha$),\ndeclination ($\\delta$), systemic velocity ($V_{sys}$), inclination ($i$) and Position Angle (PA) of these galaxies are\ngiven in Table 1. All the galaxies selected were in the inclination range of 45 to 70 degrees.  This was to ensure the availability of good\nresolution in velocity maps and HI maps, both of which were essential for the analysis \\citep{Block02,Bournaud05}. Details of the\nobservation and the preliminary data reduction are given elsewhere \\cite{Marc01}.\n\n\\begin{table*}\n\\centering\n\\noindent\n\\caption{The sample of galaxies selected for spatial lopsidedness analysis \\citep{Marc01}}\n\\begin{tabular}{@{}lcrrccc@{}}\n\\hline\n\\hline\n\\bf{Name}     &Hubble Type& \\bf{$\\alpha$}\\small(J2000) & \\bf{$\\delta$}\\small(J2000) &\\bf{$V_{sys}$}&Inclination&Position Angle\\\\\n              &\t& ~h~~m~~s~ & ~~\\hbox{$^\\circ$}~~$'$~~$''$~& (km s$^{-1}$)&($^\\circ$)&($^\\circ$) \\\\\n                                                                                                                             \n\\hline\nUGC 6446      &Sd&11~26~40.4& 53~44~48&644.3&54&200\\\\\nNGC 3726      &SBc&11~33~21.2& 47~01~45&865.6&54&194\\\\\nNGC 3893      &Sc&11~48~38.2& 48~42~39&967.2&49&352\\\\\nNGC 3949      &Sbc&11~53~41.4& 47~51~32&800.2&54&297\\\\\nNGC 3953      &SBbc&11~53~48.9& 52~19~36&1052.3&62&13\\\\\nUGC 6917      &SBd&11~56~28.8& 50~25~42&910.7&59&123\\\\\nNGC 3992      &SBbc&11~57~36.0& 53~22~28&1048.2&58&248\\\\\nUGC 6983      &SBcd&11~59~09.3& 52~42~27&1081.9&50&270\\\\\nNGC 4051      &SBbc&12~03~09.6& 44~31~53&700.3&50&311\\\\\nNGC 4088      &Sbc&12~05~34.2& 50~32~21&756.7&71&231\\\\\nNGC 4389      &SBbc&12~25~35.1& 45~41~05&718.4&50&276\\\\\n\\hline\n\\hline\\\\\n\\end{tabular}\n\\end{table*}\n\n\nFor the sake of completeness, a brief summary of the data reduction procedure is given here. As a result of observation\nof typical duration of 12 to 60 hour with WSRT, raw UV data were obtained. These data were calibrated, interactively\nflagged and Fast Fourier Transformed using the NEWSTAR software. The resulting data cubes were further processed using\nthe Groningen Image Processing SYstem (GIPSY). All the data cubes were smoothed to $30^{\\prime\\prime}\\times\n30^{\\prime\\prime}$ and continuum subtraction was carried out. The resulting cubes were used to derive the HI-surface\ndensity (Moment 0) and HI-velocity (Moment 1) maps. The typical 3$\\sigma$ column density of $10^{20}$cm$^{-2}$ was obtained for the moment 0 maps.\nThe moment 1 maps had typical velocity resolution of $\\sim 19$kms$^{-1}$. It should be emphasised that the Eridanus angular resolution of $20^{\\prime\\prime}$ ($\\sim 2.24$kpc) and velocity resolution ($\\sim 10$kms) and column density \\citep{Omar05} were comparable to that of Ursa Major.\n\n\\subsection{Optical and Near-IR Data}\n\nThe K$^\\prime$-Band and R-Band images of a few of the largest galaxies having a typical diameter of $3^{\\prime}\n-6^{\\prime}$ in the inclination range of $49^\\circ - 62^\\circ$ were sourced from the Canadian Astronomy Data\nCentre (CADC)  archives. These images were\nobtained using various telescopes and CCD cameras by Tully et al. (1996) and is kept in the archives after the initial data\nreductions like cosmic ray removal, dark subtraction, and flat fielding were carried out.\nThe typical\nresolution of the images were $~1^{\\prime\\prime}$ (R-Band) and $\\sim 2^{\\prime\\prime}$ (K-Band). These are analysed to obtain the asymmetry in the stellar \ndistribution, and compare \nthat with the HI asymmetry (Section 3.2).\n\n\\section{Harmonic Analysis}\n\n\\subsection{Harmonic Analysis of Radio Data}\n\nWe have adopted the harmonic analysis for analysing the data (Paper I). In HI,\nwhere both velocity maps and surface density maps are available, the \nanalysis technique is different from that adopted\nin optical analysis. The procedure assumes that in an ideal galaxy, HI is in pure circular motion. Hence we have,\n\n\\begin{equation}\nV(x,y)=V_{0}+V_{c}\\cos(\\phi^{\\prime})\\sin(i)+V_{r}\\sin(\\phi^{\\prime})\\sin(i)\n\\end{equation} \n\n\\noindent where $V(x,y)$ is the velocity at the rectangular coordinate $(x,y)$,$V_{0}$ is the systemic velocity, $V_{c}$ is the rotation velocity, $i$ is the\ninclination and $V_{r}$ is the expansion velocity which was taken to be zero. The azimuthal angle ($\\phi^{\\prime}$) measured in the plane\nof the galaxy, is given by the equations\n\n\\begin{equation}\n\\cos(\\phi^{\\prime}) ={\\frac {-(x-x_{0})\\sin(PA)+(y-y_{0})\\cos(PA)}{r}}\n\\end{equation}\n\\begin{equation}\n\\sin(\\phi^{\\prime})={\\frac{-(x-x_{0})\\cos(PA)+(y-y_{0})\\sin(PA)}{r\\cos(i)}}\n\\end{equation}\n\n\\noindent where $r=\\sqrt{((x-x_{0})^2+(y-y_{0})^2\/\\cos(i)^2)}$. In these equations, $(x_{0},y_{0})$ is the kinematical centre of\nthe galaxy, $PA$ is the position angle of the galaxy measured in the anti-clockwise direction from the north direction.\nUsing these equations, the five unknown parameters, namely $(x_{0},y_{0})$, PA, $V_c$ and $i$ were estimated using the GIPSY\ntask ROTCUR \\citep{Baldwin80} in an iterative manner \\citep{Wong03,Omar05}. It was observed that the dynamical centre, derived from\nvelocity maps were less than $2^{\\prime\\prime}$ away from the optical centre. \nHence, for all the calculations the optical centre was used. \n\n\\subsubsection{Spatial Lopsidedness and Other Non-axisymmetry in HI}\n\nThe harmonic coefficients were derived from the surface density maps, assuming that the surface density at\neach radii can be expanded in the form\n\n\n\\begin{equation}\nI(r,\\phi^{\\prime})= a_0(r) + \\sum_{m}a_m\\cos m[\\phi^{\\prime} - \\phi_{m}(r)]\n\\end{equation}\n\nHere, $a_m$ is the amplitude of the surface density harmonic coefficient and $\\phi_{m}(r)$ is the phase. The harmonic coefficients so\nderived were normalised using the mean surface density ($a_0$) at each radius. The variation of \nthe normalised amplitude of the first order harmonic\ncoefficient A$_1 (=a_1\/a_0)$ and of the phase angle $\\phi_1$ with respect to \nthe radius are shown (Figures 1 \\& 2).The values for the average A$_1$ measured in the larger range 1.5-2.5 $R_{K'}$ is given in column 4 of table 2. These values can be compared with the values estimated by earlier workers \\citep{Angiras06,Bournaud05,Rix95}. Similarly the values of the fractional Fourier\namplitudes A$_1$, A$_2$, and A$_3$ \ncorresponding respectively to the Fourier components m=1,2,3 in the range 1-2 R$_w$are given in Table 2 (columns 6,7 \\&)- see Section 3.3 .\n\n\\begin{figure*}\n\\includegraphics[width=84mm,height=25mm]{UGC_6446.normA1.ps}\n\\includegraphics[width=84mm,height=25mm]{NGC_3726.normA1.ps}\n\\includegraphics[width=84mm,height=25mm]{NGC_3893.normA1.ps}\n\\includegraphics[width=84mm,height=25mm]{NGC_3949.normA1.ps}\n\\includegraphics[width=84mm,height=25mm]{NGC_3953.normA1.ps}\n\\includegraphics[width=84mm,height=25mm]{UGC_6917.normA1.ps}\n\\includegraphics[width=84mm,height=25mm]{NGC_3992.normA1.ps}\n\\includegraphics[width=84mm,height=25mm]{UGC_6983.normA1.ps}\n\\includegraphics[width=84mm,height=25mm]{NGC_4051.normA1.ps}\n\\includegraphics[width=84mm,height=25mm]{NGC_4088.normA1.ps}\n\\includegraphics[width=84mm,height=25mm]{NGC_4389.normA1.ps}\n\\caption{ The asymmetry parameter derived from the surface density maps (moment 0). In each of the maps, the radius is in the units of K'-band scale length. The mean value estimated is for the complete range.}\n\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=84mm,height=25mm]{UGC_6446.normPhi1.ps}\n\\includegraphics[width=84mm,height=25mm]{NGC_3726.normPhi1.ps}\n\\includegraphics[width=84mm,height=25mm]{NGC_3893.normPhi1.ps}\n\\includegraphics[width=84mm,height=25mm]{NGC_3949.normPhi1.ps}\n\\includegraphics[width=84mm,height=25mm]{NGC_3953.normPhi1.ps}\n\\includegraphics[width=84mm,height=25mm]{UGC_6917.normPhi1.ps}\n\\includegraphics[width=84mm,height=25mm]{NGC_3992.normPhi1.ps}\n\\includegraphics[width=84mm,height=25mm]{UGC_6983.normPhi1.ps}\n\\includegraphics[width=84mm,height=25mm]{NGC_4051.normPhi1.ps}\n\\includegraphics[width=84mm,height=25mm]{NGC_4088.normPhi1.ps}\n\\includegraphics[width=84mm,height=25mm]{NGC_4389.normPhi1.ps}\n\\caption{ The asymmetry phase parameter derived from the surface density maps (moment 0). In each of the maps, the radius is in the units of K'-band scale length.}\n\n\\end{figure*}\n\nThe results from Figs. 1-2 and Table 2 can be summarised as:\n\\begin{enumerate}\n\\item{The amplitude of disk lopsidedness shows an increase with\nradius, and the phase is nearly constant with radius\nas also seen earlier for the field galaxies (Rix \\& Zaritsky\n1995), and the group galaxies (Paper I). We note, however, that these correlations are not so robust in the \nUrsa Major case. These similarities, and the differences, \nprovide important clues to the origin of the lopsidedness in the field and group galaxies.\n\nIn contrast, the advanced mergers of galaxies show an amplitude $A_1$ \nthat peaks at an intermediate radius of a few kpc and then turns over, \nand the phase\nshows a large fluctuation with radius (Jog \\& Maybhate 2006).\nThese two different properties  clearly \nunderline the different mechanism for the origin of lopsidedness\nin the present sample of normal galaxies \nas compared to the mergers of  galaxies.}\n\n\n\\item{The average value of the lopsidedness is \n$\\sim 0.14 \\pm 0.05$ (Table 2,column 4). This is similar to the mean\n value for the\nfield galaxies obtained over the same radial region of\n1.5-2.5 disk scalelengths  (Rix \\& Zaritsky 1995, Bournaud et\nal. 2005) and about half of the average value that is seen in the Eridanus group (Paper I). In addition, only 2 out of the 11 galaxies (or $\\sim 20\\%$) of this sample have A$_{1}\\ge 0.2$ and none have A$_{1}\\ge 0.3$. On the other hand, the Eridanus group of galaxies (Paper I)   showed higher values ($\\sim 40\\%$ and $\\sim 30\\%$ of the sample had A$_{1}$ values higher than 0.2 and 0.3 respectively)}.\n\nThus, despite being in a group environment, the Ursa Major galaxies\nshow overall smaller $A_1$ values; this point is highlighted in\n Figure 3 where histograms for the $A_1$ values for the Ursa\nMajor and Eridanus groups are  plotted. To verify that this is not a spurious effect due to the limited size of the samples, we have carried out Kolmogorov-Smirnov (KS) test on the data samples, including 3 values of A$_1$ estimated from R-Band analysis (see Discussion ). The D statistics value was estimated to be $0.357$. The probability that the two samples come from the same distribution is 23.6\\%.\nThis points to a different physical reason for the origin of the observed lopsidedness in this group (see Section 4 for a detailed\ndiscussion).\n\n\\item{In addition, since the HI extends much farther out and can be\nstudied to a larger radii than the stars, we have measured $A_1$\nvalues to larger radial range going up to 4-6 disc scalelengths as compared to\n2.5 disc scalelengths possible in the stellar case (Rix \\& Zaritsky 1995,\nalso see Section 3.2 in the present paper).}\n\n\\item{The values of asymmetry as measured by the Fourier amplitudes $A_1$, $A_2$, \nand $A_3$ over the range 1-2 $R_w$ (see Table 2) are comparable. In contrast, \nthe field galaxies show the amplitudes $A_1$ and A$_2$ for m=1,2 to be stronger than \n$A_3$ for m=3 in general (Rix \\& Zaritsky 1995, Bournaud et al. 2005), and in centres of advanced mergers it was found that $A_3$ is large only when $A_1$ is large, and in any case the $A_3$ values are always smaller than the $A_1$ values (Jog \\& Maybhate 2006).\nThe similar values of the amplitudes of m=1,2,3 in the Ursa Major case could be due to the complex group potential with possible  multiple interactions which may be reflected \nin the amplitudes of m=3 and higher modes.}\n\n\\end{enumerate}\n\\begin{figure*}\n\\includegraphics{UrsaMajorA1Histogram.ps}\n\\includegraphics{EridanusA1Histogram.ps}\n\\caption{ The  histograms showing the  number of galaxies vs.\n$<A_1>$ in the 1.5 to 2.5 $R_{K'}$ range for the Ursa Major Group (left) and the Eridanus Group (right) of galaxies.\nClearly, the Ursa Major galaxies show overall smaller amplitudes of lopsidedness.}\n\n\\end{figure*}\n\n\\begin{table*}\n\\centering\n\\noindent\n\\caption{ The mean values of $A_1$ in the range 1.5-2.5 $R_{K'}$, and $A_1$, $A_2$, $A_3$ in the range 1-2 $R_w$ }\n\\begin{tabular}{@{}lccccccc@{}}\n\\hline \n\\hline \n\\bf{Name}& Hubble Type&$R_{K'}$&$<A_1>_{K'}$&$R_w$&$<A_1>_w$&$<A_2>_w$&$<A_3>_w$\\\\\n\t &\t       &(kpc) &$1.5-2.5R_{K'}$&(kpc)&$1-2 R_w$&$1-2 R_w$&$1-2 R_w$\\\\\n\\hline\nUGC 6446 &7 &0.82 &0.14&3.19 &0.17&0.23& 0.08 \\\\\nNGC 3726 &5 &2.12 &0.11&4.08 &0.16&0.12&0.12\\\\\nNGC 3893 &5 &1.70 &0.20&$--$ &$--$&$--$&$--$ \\\\\nNGC 3949 &4 &1.00 &0.16&2.69 &0.22&0.27&0.07 \\\\\nNGC 3953 &4 &2.90 &0.13&3.96 &0.17&0.28&0.17 \\\\\nUGC 6917 &7 &1.90 &0.13&3.80 &0.13&0.21&0.06 \\\\\nNGC 3992 &4 &3.11 &0.23&$--$ &$--$&$--$&$--$\\\\\nUGC 6983 &6 &2.06 &0.03&4.43 &0.13&0.07&0.11 \\\\\nNGC 4051 &4 &1.37 &0.15&2.86 &0.17&0.25&0.14 \\\\\nNGC 4088 &4 &1.81 &0.08&4.13 &0.19&0.18&0.11 \\\\\nNGC 4389 &4 &0.74 &0.14&$--$ &$--$&$--$&$--$ \\\\\n\\hline\nMean\t&  &\t  &$0.14\\pm0.05$    &     &$0.17\\pm0.03$&$0.20\\pm0.07$&$0.11\\pm0.04$ \\\\\n\\hline\n\\hline\\\\\n\\end{tabular}\n\\end{table*}\n\n\\subsubsection{Kinematical Lopsidedness in HI}\n\nThe five parameters i.e. the coordinates of the centre ($x_{0},y_{0}$), systemic velocity ($V_{0}$), circular velocity ($V_{c}$), inclination ($i$) and the position angle ($PA$),  estimated from the velocity maps using the iterative use of GIPSY routine ROTCUR. These parameters were given as the input to another GIPSY routine called RESWRI along\nwith the velocity maps and the HI-surface density maps to obtain the harmonic coefficients.\n\nAt each radii (r), the line of sight velocity was expanded in the form\n\n\\begin{equation}\nv_{los}(r,\\phi^{\\prime})=c_0+\\sum_{m=1}c_m\\cos(m\\phi^{\\prime})+s_m\\sin(m\\phi^{\\prime})\n\\end{equation}\n\nwhere, $c_m$,$s_m$ are the harmonic coefficients, $c_0$ is identical to the systemic velocity $V_{0}$ and $\\phi^{\\prime}$ is the azimuthal angle. These\nharmonic coefficients were derived at concentric radii which were separated by $15^{\\prime\\prime}$.In our analysis we have derived the harmonic coefficients up to the $10^{th}$ order. This was partially prompted by the observation that effects of  bars tend to retain the strength of the Fourier coefficients even for m=10 terms \\citep{Buta03}. Typical velocity harmonic\ncoefficients are shown in Figure 4.\n\n\\begin{figure*}\n\\includegraphics[width=185mm,height=165mm]{NGC_3726.harmon.ps}\n\n\\caption{Velocity harmonic coefficients estimated for NGC 3726. The power in each of the harmonic order is shown in the bottom left hand panel. The bottom right hand panel shows the $c_0$ coefficient ($V_{sys}$) as a function of radius.}\n\n\\end{figure*}\n\nFrom these coefficients, since $c_3 \\sim 0$, it is seen that the \ninclination fitting has converged \\citep{Schoen97}. In addition, an estimate of\nthe effects of spiral arms and global elongation in the potential \nthat gives rise to the kinematical lopsidedness can be obtained from \nthe $s_1$ \\& $s_3$ coefficients \\citep{Schoen97}. If the influence \nof spiral arms are large, $s_1$ and $s_3$ are expected to\noscillate rapidly \\citep{Schoen97}. Since this is not seen in Figure 4, their contribution\nmust be small. \n \nFrom the harmonic coefficients thus obtained, it is possible to estimate the \nelongation in the potential of the galaxy times\n a factor of $\\sin(2\\phi_2)$, where $\\phi_2$ is the \nphase angle for $m=2$ in the plane of the galaxy \\citep{Schoen97}. \nThe estimation of the ellipticity of the potential of an early type galaxy, \nIC 2006, using the velocity field for the HI ring in it was first\ncarried out by Franx et al. (1994). This was later generalised to include \nspiral galaxies with extended exponential disks \\citep{Schoen97}. In this\nprocedure, with the assumptions of flat rotation curve in the outer regions of the galaxy and constant\nphase ($\\phi_2 (r) $), the ellipticity of potential ($\\epsilon_{pot}$) is obtained as\n$\\epsilon_{2}\\sin(2\\phi_{2}(r))$ \\citep{Schoen97}.\nWe have carried out similar analysis for all \nthe sample galaxies in the Ursa Major group, and presented\nin Section 3.3.\nA typical asymmetry or the elongation in the potential for NGC 3726 is shown \nin Figure 5.\n\n\\begin{figure*}\n\\includegraphics[width=110mm,height=70mm]{NGC_3726.potellip.ps}\n\\caption{ The estimated elongation in potential $\\epsilon_{pot}\n= \\epsilon_2\\sin(2\\phi)$ derived from \nthe velocity harmonic coefficients for NGC 3726}\n\\end{figure*}\n\n\\subsection{Harmonic Analysis of Optical and Near-IR Data}\nThe harmonic analysis of the optical (R-band and K'-band) data was carried out as per \nthe procedure adopted by \nearlier workers \\citep{Rix95, Zaritsky97, Angiras06}. \nThe original images obtained from CADC were corrected for the sky background \nand for the atmospheric extinction. In addition to this, foreground stars were masked.\nThe optical centres of the galaxies were estimated using the IRAF\ntask IMCNTR. It was seen that the optical centres of these galaxies were the same as that obtained by Tully et al. (1996). These images were\ndeprojected using the IRAF {\\footnote {IRAF is distributed by the National Optical Astronomy Observatories,\nwhich are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with\nthe National Science Foundation.}} task IMLINTRAN \\citep{Buta98}. In this deprojection, we have not taken into account\nthe effects of the bulge of the galaxy. The effect of bulge is expected to be very small as we are mainly interested in\nthe outer regions of the galaxy \\citep{Bournaud05}. In addition we expect bulge contamination to be serious for the m=2 mode and for almost edge-on galaxies which is not relevant in our case. For each of the galaxies, along various concentric annuli, the surface density as a function of angle was\nextracted using the ELLIPSE {\\footnote {ELLIPSE is a product of the Space Telescope Science Institute, which is\noperated by AURA for NASA.}}task. Each of the rings were separated by $1^{\\prime\\prime}$ (typical resolution) in the case of R-Band images. Harmonic analysis was carried out on the extracted\nsurface density values and normalised coefficients were estimated. The variation of $A_1$ coefficients,\nderived from R-Band image along with those derived from the HI analysis for \ntwo sample galaxies NGC 3726, NGC 4051 are shown \nin Figure 6.\n\n\\begin{figure*}\n\\includegraphics[width=84mm,angle=-90]{N3726RA1.ps}\n\\includegraphics[width=84mm,angle=-90]{N4051RA1.ps}\n\\caption{The $A_1$ coefficients derived from R-Band images of galaxies, \nwhich are compared\nwith that obtained from HI surface density maps. Note that the values are comparable in the inner regions \nof radial overlap.}\n\n\\end{figure*}\n\nFrom this figure, it can be seen that the lopsidedness in the\nstellar distribution measured \nfrom the optical data is comparable to that\nseen in HI in the same radial region of study. \nThus these values represent true lopsidedness and not features seen only for the\nkind of tracer used. This agreement confirms what\nwas also seen in our earlier study of HI lopsidedness in\nEridanus (Paper I), and provides evidence that the\nasymmetry in both arise due to the stars and gas responding respectively to\nthe same perturbation potential as proposed by Jog (1997). \nIn the outer parts, only HI is available as a tracer \nsince the near-IR data is available only  up to 2.5 disc \nscalelengths (see Section 3.1.1).\n\n\n\\subsection{Strength of the Perturbation Potential}\n\nAssuming that the asymmetry arises due to the disc response to a\ndistorted halo (Jog 1997), from the surface density maps, we can obtain the\nperturbation potential corresponding to the m=1,2,3 terms from\nthe observed amplitudes $A_1$, $A_2$ and $A_3$ for the normalised Fourier coefficients (Jog 2000). Here the perturbation potential is taken to be of\nthe form ${V_c}^2 \\epsilon_m  cos m\\phi $ where $V_c$ is the\nflat rotation curve value and $\\epsilon_m$ denotes the\nperturbation parameter. As a result, it can be shown that $\\epsilon_1$ denotes the lopsided potential,\nand $\\epsilon_2$ denotes the elongation or ellipticity of the\nperturbation potential.\n\nThe observed HI distribution was fitted with a Gaussian curve and the associated {\\it Gaussian scalelength} $R_w$ was estimated. Using this scalelength the relations between $\\epsilon_m$, $A_m$ and $R_w$ have been derived for m = 1, 2 and 3, following the procedure in Jog (2000) where it was developed for an exponential disk distribution\nin a region of flat rotation curve. It is now believed that haloes of galaxies in groups are merged in the outer parts \\citep{Athanassoula97}, however in the inner regions of 10 kpc where we study the asymmetry, we can still treat the asymmetry as if the halo were isolated. Hence the above model of disk response to a halo is still reasonably valid and the inner regions could carry the signatures of complex tidal interactions in a group setting.\nThis yields the following relations between the perturbation\nparameters for the potential $\\epsilon_m$ and the $A_m$ values:\n\n\\begin{equation}\n\\epsilon_2 = \\frac {A_2(r)} {(r\/R_w)^2 + 1} \n\\end{equation}\n\nand\n\n\\begin{equation}\n\\epsilon_3 = \\frac {A_3(r)} {(2\/7){(r\/R_w)^2} + 1}\n\\end{equation}\n\nThe relation for m=1 was already obtained in Paper I,\nand is:\n\n\\begin{equation}\n\\epsilon_1 = \\frac {A_1(r)}  {2(r\/R_w)^2 - 1}\n\\end{equation}\n\n\nThe resulting mean values of the perturbation parameters are obtained using the measured values of the Fourier amplitudes $A_1$, $A_2$ and $A_3$, and are given in Table 3.\n\nUsing the analysis of the kinematical data, we have also obtained\n$\\epsilon_{2} \\sin [2 \\phi_2 (r)]$ for the galaxies in Ursa Major\nas discussed in Section 3.1.1. The mean values of this quantity in the range 1-2 $R_w$ are shown in \nTable 3.\n\nThe main results from this subsection are:\n\\begin{enumerate}\n\\item{The average value of $\\epsilon_1$ or the lopsided\nperturbation for the\npotential  obtained in the outer parts (in the radial range 1-2 $R_w$) is \n$\\sim 6 \\%$, this is smaller than the value for  \nthe Eridanus case where the halo lopsided potential was derived to be $10 \\%$ - this reflects the smaller observed amplitudes of lopsidedness in the present sample.\n\nThus if the lopsidedness is due to the response to the halo\ndistortion, then this gives 6\\% as the typical halo\nlopsidedness for the Ursa Major galaxies.}\n\n\n\\item{The elongation in the potential or the magnitude or the amplitude of the elongation in the term $\\epsilon_2$ value is comparable (within a factor of $\\sin(2\\phi_2)$)\nwhether calculated from the observed spatial asymmetry or from the kinematical asymmetry (see columns 4 and 6 of Table 3). This confirms the argument (Jog 1997, Jog 2002) that both spatial and kinematical asymmetry\nresult from the same perturbation potential.}\n\n\\item{The values of all three perturbation potentials derived $\\epsilon_1, \\epsilon_2 , \\epsilon_3$  are\ncomparable. Although this result depends on the model used, it reinforces the similar result obtained for the Fourier amplitudes which are directly observed and hence are model-independent (Section 3.1.1).  This can be an important clue to the mechanism for\ngenerating lopsidedness in groups, and perhaps indicates the importance of  multiple simultaneous tidal interactions  that can occur under the special conditions of a group environment.}\n\\end{enumerate}\n\n\\begin{table*}\n\\centering\n\\noindent\n\\caption{The HI-scalelength, and the mean perturbation parameters of potentials obtained from $A_1$, $A_2$ and $A_3$- the coefficients of surface densities and from velocity fields. The mean values are calculated between 1-2 $R_w$ }\n\\begin{tabular}{@{}lccccc@{}}\n\\hline\n\\hline\n\\bf{Name}&HI Scalelength ($R_w$) (kpc)&$<\\epsilon_{1}>$&$<\\epsilon_{2}>$&$<\\epsilon_{3}>$&$<\\epsilon_{2}\\sin(2\\phi_2)>$\\\\\n\\hline\nUGC 6446       &3.19     &0.046      &0.040     & 0.065     &-0.173\\\\\nNGC 3726       &4.08     &0.049      &0.042     & 0.111     &-0.392\\\\\nNGC 3949       &2.69     &0.072      &0.106     & 0.056     & 0.178\\\\\nNGC 3953       &3.96     &0.044      &0.098     & 0.113     &-0.171\\\\\nUGC 6917       &3.80     &0.041      &0.057     & 0.083     & 0.007\\\\\nUGC 6983       &4.43     &0.032      &0.022     & 0.128     &-0.213\\\\\nNGC 4051       &2.86     &0.082      &0.113     & 0.109     &-0.087\\\\\nNGC 4088       &4.13     &0.089      &0.076     & 0.092     & 0.001\\\\\n\\hline\\\\\nMean\t       &\t &$0.057\\pm0.021$&$0.069\\pm0.034$&$0.095\\pm0.025$&$-0.106\\pm0.172$\\\\\n\\hline\n\\hline\\\\\n\\end{tabular}\n\\end{table*}\n\n\\section{Discussion : Lopsidedness in groups}\n\\begin{enumerate}\n\\item {The Ursa Major group of galaxies show a typical lopsidedness of $\\sim\n14 \\% $ in the\ninner regions, that is comparable to the field case, and about\nhalf of what is seen in the Eridanus group (see Section 3.1.1\nfor details).\n\nWe also measure the $A_1$ values in the outer parts between 1-2\n$R_w$, and find this to be $\\sim 17 \\%$. Again this is smaller by a\nfactor of $\\sim 1.6$ compared to the Eridanus case. }\n\n\\item{We plot  the mean $A_1$ ($<A_1>$) value in the inner regions of the \ngalaxies with respect to Hubble type in Figure 7, where we also\nplot the corresponding values from the Eridanus study for comparison (Paper I, Figure 5) .\nNote that the early-type galaxies in the Eridanus group show a higher lopsidedness, this\nis opposite to what is seen in the field galaxies \\citep{Zaritsky97,Bournaud05}, and points to tidal interactions as the mechanism for the origin of lopsidedness as argued in Paper I. In contrast, in field galaxies, gas accretion plays an important role in generating lopsidedness \\citep{Bournaud05}. \nThe anti-correlation with galaxy Hubble type is weaker in the\nUrsa group case, perhaps because our sample here only covers a smaller\nsubgroup of galaxy types from type 4 to 7, whereas the Eridanus\nstudy spans a much larger range from type 1 to 9.\n\nTo address this issue, we obtained the R-Band images of three more galaxies from the Ursa Major sample \\citep{Tully96}, namely UGC 6930, NGC 4102 and NGC 3729, from the Canadian Astronomy Data Centre (CADC) on which Fourier analysis was carried out. UGC 6930 belonged to Hubble type 7 and had an inclination of 32$^\\circ$. NGC 4102 and NGC 3729 belonged to Hubble type 2 and had inclinations 58$^\\circ$ and 48$^\\circ$ respectively. The A$_1$ coefficient in the range 1.5 to 2.5 $R_{K'}$ for UGC 6930 was estimated to be 0.02. In the same range, the A$_1$ coefficients for NGC 4102 and NGC3729 were 0.12 and 0.04 respectively. These galaxies were not included in our HI analysis because of lack of reliable HI data. It should be noted that if these three points are included in KS-test, the probability that the values of A$_1$ come from the same distribution is $23.6\\%$ while the maximum difference (D) between the cumulative distribution is 0.357.    \n The three resulting points are shown in Fig. 7 (denoted by symbol \\mbox{$\\triangle\\hspace{-0.072 in}\\cdot$\\hspace{0.072 in}}) and they confirm that\n the distribution of A$_1$ vs. R is nearly flat. Thus there is no clear anti-correlation in this case unlike that seen in the Eridanus group. However, this distribution does not show a positive correlation with the Hubble type either, unlike the field case \\citep{Bournaud05}.} Thus Figure 7 confirms the different physical origins for the lopsidedness in the group and field cases. It also confirms that the anti-correlation seen in the Eridanus group can be attributed to the group environment, and requires a higher galaxy number density as seen in the Eridanus to be effective.\n\n\\item {The kinematical analysis gives a value for the elongation in\nthe potential, showing all galaxies where such analysis could be\ncarried out to be disturbed. This was found earlier for a group\nof five galaxies in the Sculptor group galaxies (Schoenmakers 2000).}\n\n\\item {The above results show that the group environment is\nconducive to producing lopsidedness, with tidal interactions playing a major role in this. Galaxy interactions can give rise both to lopsidedness and a secular evolution towards\nearly-type galaxies as argued by Bournaud et al. (2005).\nThere are indeed some indications of tidal interactions in Ursa Major \ngroup of galaxies (Verheijen \\& Sancisi 2001).}\n\\end{enumerate}\n\n\\begin{figure*}\n\\includegraphics[width=84mm,angle=-90]{A1VsHubTypeRKnofitNew.ps}\n\\caption{The $<A_1>$ values in the 1.5 to 2.5 K$^\\prime$-band scale length of Ursa Major galaxies. The Eridanus Group\nvalues are taken from Paper I.  The values denoted by \\mbox{$\\triangle\\hspace{-0.072 in}\\cdot$\\hspace{0.072 in}}, correspond to A$_1$ values of the 3 galaxies estimated from R-Band analysis (see Discussion ).The A$_1$ values are higher for the early-type galaxies in the Eridanus while the distribution is flatter for the Ursa Major group.}\n\\end{figure*}\n\nThe Ursa Major group is a loose group, and has number density that is intermediate between the field and the Eridanus values. Hence tidal interactions are less important in the Ursa Major group, which could explain the lower amplitude of lopsidedness (A$_1$).\n\nThe lower values of asymmetry parameter ($<A_1>$) observed for Ursa Major group of galaxies may also find a partial\nexplanation, if we assume that it mainly falls on a filament and is in the process of forming a group. Such a process\nis observed in ZwCl 2341.1+0000 \\citep{Bagchi02}.\n\n\\section{Conclusions}\n\nThe main conclusions drawn from this paper are as follows:\n\\begin{enumerate}\n\\item{The mean amplitude of disk lopsidedness in the Ursa Major group galaxies is \nmeasured and found to be comparable to the field sample, while the \nEridanus group showed a factor of two higher lopsidedness. \n\nThe smaller\namplitudes of lopsidedness seen in the present study\ncould be due to the lower galaxy number density and the lower velocity\ndispersion in the Ursa Major group (see Section 2). The group environment and tidal interactions are shown to play a major role in generating lopsidedness, especially in a denser group like the Eridanus. \n\nThe disk lopsidedness can thus be used as diagnostics to study the\ngalaxy interactions and the halo properties in groups of\ngalaxies.}\n\n\\item{The values of elongation of potential as measured from\nspatial and kinematical studies gives comparable values, thus\nsupporting the idea  (Jog 1997) that both types of asymmetry arise due to\nthe same perturbation potential.}\n\n\\item{The values of the asymmetry as measured by the mean fractional Fourier amplitudes A$_1$, A$_2$ and A$_3$ are found to be comparable,\nand also the derived perturbation potential parameters $\\epsilon_1$, $\\epsilon_2$ and $\\epsilon_3$ are found to be comparable. This is in\ncontrast to the field galaxies where A$_1$ and A$_2$ are stronger than A$_3$ and higher mode amplitudes. This indicates the importance of multiple tidal interactions that can occur under the special conditions of a group environment.}\n\\end{enumerate}\n\n\n\n\\section{ACKNOWLEDGMENTS}\n\nWe thank the referee, Frederic Bournaud, for a careful reading of the manuscript \nand for the critical comments and the suggestion of including the A$_1$ values from R-Band images in Figure 7. These have improved the presentation of the paper.\nRAA takes great pleasure in thanking K. Indulekha, School of Pure and Applied Physics, M.G.University,for her constant encouragement during this project. He also thanks the University Grants Commission of India and St.Joseph's College, Bangalore for granting study leave under the FIP leave of 10th five year plan and Raman Research Institute, Bangalore for providing all the facilities to pursue this study.This research used the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDeviations from the exponential decay law of unstable systems are a natural\nconsequence of the postulates of Quantum Mechanics\n\\cite{1978RPPh...41..587F,khalfin,2008JPhA...41W3001F}: for an unstable state,\nwhose average energy is finite, the survival probability for short times after\nthe \\textquotedblleft creation\\textquotedblright\\ of the state is slower than\nany exponential decay law. In other terms, if we introduce an effective, time\ndependent decay rate $\\gamma(t)=\\frac{-1}{t}\\mathrm{log}(p(t))$ one has that\nfor $t\\rightarrow0^{+}$, $\\gamma(0^{+})=0$ while at large times the standard\nexponential decay law, $\\gamma(t)\\simeq\\Gamma$, is obtained. The initial\ntemporal window for which deviations from the exponential law take place is\nusually very small: it is of the order of $10^{-15}$ sec for electromagnetic\natomic decays \\cite{Facchi:1999nq}. This explains why these deviations have\nnever been observed in experiments before 1997 \\cite{raizen1} when, for the\nfirst time, a cold atoms experiment has reported the evidence of such\ndeviations for \\textquotedblleft bona fide\\textquotedblright\\ unstable states\n(tunneling of atoms out of a trap). Previously, in \\cite{1990PhRvA..41.2295I},\ndeviations from the exponential law have been reported within Rabi\noscillations. The short time deviations from the exponential law open up the\npossibility of the so called Quantum Zeno effect \\cite{1977JMP....18..756M,2008JPhA...41W3001F,Facchi:1999nq}:\nby \\textquotedblleft observing\\textquotedblright\\ the system with pulsed\nmeasurements at short times after its preparation, the effective decay rate is\nreduced and eventually it vanishes for continuous measurements (Quantum Zeno\nparadox). Also this prediction has been recently confirmed within cold atoms\nexperiments \\cite{raizen2} and, moreover, the so called Inverse Quantum Zeno\neffect has also been observed: in this case the measuring apparatus leads to a\nfaster decay of the unstable state \\cite{2000Natur.405..546K}.\n\nA natural question concerns the existence of deviations from the exponential\nlaw also in the context of Relativistic Quantum Field Theory (RQFT) which is\nthe right theoretical frame for describing unstable particles. In the\nperturbative approaches presented in \\cite{1993PhRvL..71.2687B} no (or very\nmuch suppressed) short-time deviations from the exponential law, and thus no\nquantum Zeno effect, were found within RQFT. Here and in\nRef.~\\cite{Giacosa:2010br} we reconsider the issue of the survival probability\nin RQFT also by analyzing some subtleties one faces when trying to define\nunstable particles, such as the problem of \\textquotedblleft preparation of\nthe system\\textquotedblright\\ and of the fields redefinition. We will not\nconsider here the case of the fundamental Lagrangian of the Standard model but\nwe limit the discussion to a toy model superrenormalizable Lagrangian. We\nindeed find that deviations from the exponential law occur also in a genuine\nRQFT context and we discuss possible implications for hadronic decays.\n\n\\section{A model Lagrangian}\n\nThe toy Lagrangian we use to investigate the survival probability of an\nunstable scalar particle $S$ decaying into two scalars $\\varphi$ is given by:\n\\begin{equation}\n\\mathcal{L}=\\frac{1}{2}(\\partial_{\\mu}S)^{2}-\\frac{1}{2}M_{0}^{2}S^{2}%\n+\\frac{1}{2}(\\partial_{\\mu}\\varphi)^{2}-\\frac{1}{2}m^{2}\\varphi^{2}%\n+gS\\varphi^{2}.\\label{lag}%\n\\end{equation}\nThe interaction term $\\mathcal{L}_{int}=gS\\varphi^{2}$ is responsible for the\ndecay process $S\\rightarrow\\varphi\\varphi$, whose tree-level decay rate reads:\n\\begin{equation}\n\\Gamma_{S\\varphi\\varphi}^{\\text{t-l}}=\\frac{\\sqrt{\\frac{M_{0}^{2}}{4}-m^{2}}%\n}{8\\pi M_{0}^{2}}(\\sqrt{2}g)^{2}\\text{ .}\\label{tl1}%\n\\end{equation}\nThe `naive', tree-level expression of the survival probability $p(t)$ for the\nresonance $S$ created at $t=0$ is $p_{\\text{t-l}}(t)=e^{-\\Gamma_{S\\varphi\n\\varphi}^{\\text{t-l}}t}$ and the tree-level expression of the mean life time\nis $\\tau_{\\text{t-l}}=1\/\\Gamma_{S\\varphi\\varphi}^{\\text{t-l}}$. Here we\ninterpret our Lagrangian as an effective model to describe the decays of\nhadrons; it is therefore quite natural to introduce a cutoff $\\Lambda$ on the\nenergy of the particles of the typical mass scale of strongly interacting\nparticles i.e. $\\Lambda\\sim1$ GeV. To introduce the cutoff in a more\nconsistent way one has to insert a nonlocal interaction in the Lagrangian\n\\cite{Giacosa:2007bn}:%\n\n\\begin{equation}\n\\mathcal{L}=\\frac{1}{2}(\\partial_{\\mu}S)^{2}-\\frac{1}{2}M_{0}^{2}S^{2}%\n+\\frac{1}{2}(\\partial_{\\mu}\\varphi)^{2}-\\frac{1}{2}m^{2}\\varphi^{2}%\n+\\mathcal{L}_{int}\\text{ ,}%\n\\end{equation}%\n\\begin{equation}\n\\mathcal{L}_{int}=gS(x)\\int\\mathrm{d}^{4}\\mathrm{y}\\varphi(x+y\/2)\\varphi\n(x-y\/2)\\Phi(y)\\text{ ,}%\n\\end{equation}\nwhere $\\Phi$ is a form factor whose Fourier transform, $f_{\\Lambda}(q)=\\int\nd^{4}y\\Phi(y)e^{-iyq}$, appears in the loop integrals and regularizes the\ndivergences. (For nonlocal Lagrangians see also Refs. \\cite{nl} and refs.\ntherein.) In this work we will consider the case of a sharp cutoff and the\ncase of a smooth form factor. An intermediate step to obtain the survival\nprobability is the computation of the self energy which reads:\n\\begin{equation}\n\\Sigma(x=\\sqrt{p^{2}},m)=-i\\int\\frac{d^{4}q}{(2\\pi)^{4}}\\frac{f_{\\Lambda\n}(q^{0},\\overrightarrow{q})^{2}}{\\left[  (q+p\/2)^{2}-m^{2}+i\\varepsilon\n\\right]  \\left[  (-q+p\/2)^{2}-m^{2}+i\\varepsilon\\right]  }%\n\\end{equation}\nand modifies the propagator $\\Delta_{S}$ of the unstable particle as usual:\n\\begin{equation}\n\\Delta_{S}(p^{2})=\\left[  p^{2}-M_{0}^{2}+(\\sqrt{2}g)^{2}\\Sigma(p^{2}%\n)+i\\varepsilon\\text{ }\\right]  ^{-1}\\text{.}%\n\\end{equation}\n\n\n\\section{Spectral functions and survival probabilities}\n\nSimilarly to the standard derivation within Quantum Mechanics, also in RQFT,\nthe survival probability can be obtained by projecting the initial unstable\nstate onto the energy eigenstates. In turn, this corresponds to the\ncalculation of the spectral function $d_{S}(x)$ of the scalar field $S$ which\nis proportional to the imaginary part of the propagator:%\n\\begin{equation}\nd_{S}(x=\\sqrt{p^{2}})=\\frac{2x}{\\pi}\\left\\vert \\lim_{\\varepsilon\\rightarrow\n0}\\mathrm{Im}[\\Delta_{S}(p^{2})]\\right\\vert \\text{ .}%\n\\end{equation}\nThe quantity $d_{S}(x)dx$ represents the probability that, in its rest frame,\nthe state $S$ has a mass between $x$ and $x+dx.$ It is correctly normalized\nfor each $g$, $\\int_{0}^{\\infty}d_{S}(x)dx=1$ and reproduces the limit\n$d_{S}(x)=\\delta(x-M_{0})$ for $g\\rightarrow0$\n\\cite{Achasov:2004uq,Giacosa:2007bn}. Notice that there are situations in\nwhich the the spectral function can be directly pin down by data because the\nthe background is small and well understood: the decay $\\phi\\rightarrow\n\\gamma\\pi^{0}\\pi^{0}$ through the intermediate $a_{0}(980)$ and $f_{0}(980)$\nmesons, the similar decay of the $j\/\\psi$ charmonium, or the hadronic decay of\nthe $\\tau$ lepton into $\\nu\\pi\\pi,$ dominated by the $\\rho$ meson for an\ninvariant $\\pi\\pi$ mass close to $\\rho$ mass (e.g. \\cite{Giacosa:2008st}).\n\nThe probability amplitude $a(t)$ and the survival probability $p(t)$ can be\nthen expressed as\n\\begin{equation}\na(t)=\\int_{-\\infty}^{+\\infty}\\mathrm{dx}\\,\\,d_{S}(x)e^{-ixt}\\text{ ,\n}p(t)=\\left\\vert a(t)\\right\\vert ^{2}\\text{ .} \\label{p(t)}%\n\\end{equation}\nThe condition $p(0)=1$ is fulfilled in virtue of the normalization of\n$d_{S}(x)$.\n\nLet us now study the first derivative of $p(t)$. We obtain that $p^{\\prime\n}(t=0)=0$ as a consequence of the fact that the integral $\\int_{0}^{\\infty\n}x\\,d_{S}(x)dx$ is finite and real (it is the mean mass $\\left\\langle\nM\\right\\rangle $, a reasonable definition for the mass of a resonance\n\\cite{Giacosa:2007bn}). This, in turn, implies that the function\n$\\gamma(t)=\\frac{-1}{t}\\ln p(t)$ vanishes for $t\\rightarrow0^{+}$:%\n\\begin{equation}\n\\lim_{t\\rightarrow0^{+}}\\gamma(t)=-\\lim_{t\\rightarrow0^{+}}\\frac{p^{\\prime\n}(t)}{p(t)}=0.\n\\end{equation}\nWe can therefore conclude that the quantum Zeno effect is perfectly possible\nin the present RQFT context.\n\n\\begin{figure}[ptb]\n\\begin{centering}\n\\epsfig{file=fig1.epsi,height=4.9cm,width=9cm}\n\\caption{Survival probability as a function of time. The solid line corresponds to the choice of a sharp cutoff, the thick gray line\nto a smooth form factor, the dashed line to the exponential decay law and the dotted black and gray lines\nare the differences between the survival probability as calculated from the spectral function and the exponential decay law. The deviations from the exponential law are quite sizable at short times.}\n\\end{centering}\n\\end{figure}\n\nWe show in Fig.~1 the survival probability for the case of a sharp cutoff\n(solid line) a smooth form factor $f_{\\Lambda}(q)=1\/(1+(q\/\\Lambda)^{2})$\n(thick gray line) and the standard exponential decay law (dashed line), here\n$\\Lambda=1.5$ GeV, $M_{0}=1$ GeV, $m=m_{\\pi}$ and the tree level mean life\ntime $\\tau_{t-l}=3.27$ GeV$^{-1}$ (this fixes $g$ in the two cases). Also\ndisplayed are the differences between the survival probability as calculated\nat one loop level and the tree level exponential decay law (dotted black and\ngray lines). Notice that the time interval for which sizable deviations from\nthe exponential decay law occur is of the same order of magnitude of the mean\nlife time of the particle. This is an intriguing consequence of having\nstrongly interacting particles and could in principle lead to observable\neffects, for instance in heavy ions collisions experiments. Moreover, the\ndifference between the sharp and smooth cutoff is very small: this fact\nensures that our results depend only slightly from the form of the cutoff function.\n\n\\section{Discussion and Conclusions}\n\nThere is an important issue that must be considered in connection with the\nmeasurability of these deviations, which also correspond to the measurability\nof the spectral function.\n\nFirst, we notice that for very broad resonances, for which the deviation from\nthe exponential law are strong, one should also consider the mechanism by\nwhich these resonances are created as, for instance, the scattering\n$\\varphi\\varphi\\rightarrow S\\rightarrow\\varphi\\varphi$ \\cite{Maiani:1997pd}.\nOne should introduce wave packets, with proper initial conditions, which\nsubstantially overlap at $t=0$. In the framework of plane waves, the full\nstate of the system can be expressed in terms of the eigenstates of the\nHamiltonian $H_{0}$:\n\\begin{equation}\n\\left\\vert s(t)\\right\\rangle =\\sum_{\\mathbf{k}}c_{\\mathbf{k}}(t)\\left\\vert\n\\varphi_{\\mathbf{k}}\\varphi_{\\mathbf{-k}}\\right\\rangle +c_{S}(t)\\left\\vert\nS\\right\\rangle .\\nonumber\n\\end{equation}\nThe coefficient $c_{S}(t)$ is vanishingly small for $t<<0$ and only for\n$t\\simeq0$ it becomes significant. If it were possible to tune the starting\nconditions in such a way that $c_{S}(0)=1,$ we would have $\\left\\vert\ns(t=0)\\right\\rangle =\\left\\vert S\\right\\rangle $ and the survival probability\nof the resonance would be exactly the one presented in the previous section.\nHowever, in general the state at $t=0$ is a superposition:\n\\begin{equation}\n\\left\\vert s(0)\\right\\rangle =\\sum_{\\mathbf{k}}c_{\\mathbf{k}}(0)\\left\\vert\n\\varphi_{\\mathbf{k}}\\varphi_{\\mathbf{-k}}\\right\\rangle +c_{S}(0)\\left\\vert\nS\\right\\rangle .\\nonumber\n\\end{equation}\nFurther evolution implies:\n\\begin{align}\n&  e^{-iHt}\\left\\vert s(0)\\right\\rangle =\\nonumber\\\\\n&  \\sum_{\\mathbf{k}}c_{\\mathbf{k}}(0)e^{-iHt}\\left\\vert \\varphi_{\\mathbf{k}%\n}\\varphi_{\\mathbf{-k}}\\right\\rangle +c_{S}(0)e^{-iHt}\\left\\vert S\\right\\rangle\n=\\nonumber\\\\\n&  \\sum_{\\mathbf{k}}c_{\\mathbf{k}}(0)e^{-iHt}\\left\\vert \\varphi_{\\mathbf{k}%\n}\\varphi_{\\mathbf{-k}}\\right\\rangle +c_{S}(0)\\left(  a(t)\\left\\vert\nS\\right\\rangle +\\left\\vert \\varphi\\varphi\\right\\rangle \\right)  .\\nonumber\n\\end{align}\nThe amplitude $a(t)$ enters in a more general expression but it is not clear a\npriori if the deviations from the exponential decay law are smeared out, in\nthe final \\textquotedblleft measurement\\textquotedblright\\ of the decay\nproducts, or if they could provide significant effects. A careful study would\nbe needed. Moreover, we plan also to investigate if the deviations from the\nexponential decay law could indeed lead to observable effects also in Particle\nPhysics experiments.\n\n\nThe work of G.~P. is supported by the Deutsche Forschungsgemeinschaft (DFG)\nunder Grant No. PA 1780\/2-1.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}