---
abstract: 'In this paper we prove that a complete minimal surface immersed in ${\mathbb{H}^2} \times {\mathbb R},$ with finite total curvature and two ends, each one asymptotic to a vertical geodesic plane, must be a model surface.  Moreover, we give a geometric description of minimal ends of finite total curvature in ${\mathbb{H}^2} \times {\mathbb R}$.'
address:
- 'Université Paris-Est Marne-la-Vallée Laboratoire d’Analyse et Mathématiques Appliquées Cité Descartes 5 Boulevard Descartes Champs-sur-Marne 77454 Marne-la-Vallée Cedex 2 France'
- 'Dipartimento di Matematica Universitá di L’Aquila via Vetoio - Loc.  Coppito 67010 (L’Aquila) Italy'
- 'Departamento de Matemática Pontifícia Universidade Católica do Rio de JaneiroRio de Janeiro 22453-900 RJ Brazil '
- 'Université Paris Diderot - Paris 7 Institut de Mathématiques de Jussieu, UMR CNRS 7586 UFR de Mathématiques, Case 7012 Bâtiment Chevaleret 75205 Paris Cedex 13 France'
author:
- 'Laurent Hauswirth, Barbara Nelli, Ricardo Sa Earp, Eric Toubiana'
title: |
    A Schoen theorem for minimal surfaces in ${\mathbb H}^2\times {\mathbb
    R}$
---

[^1]

Introduction
============

In the early eighties, R. Schoen [@Schoen] proved a beautiful theorem about minimal surfaces in Euclidean space.  Namely, a complete and connected minimal surface immersed in ${\mathbb R}^3$ with two embedded ends of finite total curvature is a catenoid. 

In his article, R. Schoen described the structure of finite total curvature ends minimally embedded in ${\mathbb R}^3,$ relying on the results of A. Huber [@Hub] and R. Osserman [@Osserman] about the Weierstrass representation of such ends. 

At the beginning of this century, the discovery of a generalized Hopf differential by U. Abresch and H. Rosenberg [@Abresch-Rosenberg] stimulated the study of minimal surfaces in three-dimensional homogeneous manifolds.  Many new embedded and complete minimal surfaces have been found in ${\mathbb{H}^2} \times {\mathbb R}$. In particular J. Pyo [@P] and F. Morabito and M. Rodriguez [@MR] have constructed, independently, a family of minimal embedded annuli with finite total curvature.  Each end of such annuli is asymptotic to a vertical geodesic plane.  We call each of such surfaces a [*two ends model surface*]{}. 

In this article we prove the following theorem. 

[**Main Theorem.**]{} [* A complete and connected minimal surface immersed in ${\mathbb{H}^2}
\times {\mathbb R}$ with finite total curvature and two ends, each one asymptotic to a vertical geodesic plane, is a two ends model surface*]{}. 

Following the same spirit of Schoen’s work, we describe the full geometry of minimal ends of finite total curvature in ${\mathbb{H}^2} \times {\mathbb R}.$ The study of such ends was first developed by the first author and H. Rosenberg in [@HR]. 

The paper is organized as follows. 

In Section \[Sec. End\], we study the geometry of minimal ends of finite total curvature.  The main geometric property is that horizontal sections of finite total curvature ends converge towards a horizontal geodesic.  In Section \[Sec. Surface\], we prove the Main Theorem.  In the Appendix, we study the geometry of curves with bounded curvature in the hyperbolic plane. 

Acknowledgements {#acknowledgements .unnumbered}
----------------

The second and the fourth authors wish to thank [*Departamento de Matemática da PUC-Rio*]{} for the kind hospitality.  The second and third author wish to thank [*Laboratoire*]{} [*Géométrie et Dynamique de l’Institut de Mathématiques de Jussieu*]{} for the kind hospitality. 

Minimal ends with finite total curvature in ${\mathbb{H}^2} \times
{\mathbb R}$ {#Sec. End}
==================================================================

In this section we give the geometrical structure of a finite total curvature end.  We rely on the complex analysis involved in the theory of minimal surfaces [@HST], [@HR], [@ST] and on the theory of harmonic maps developed by Z. Han, L. Tan, A. Treiberg and T. Wan [@HTTW] and Y. N. Minsky in [@Minsky]. 

Let $M$ be a Riemann surface and let $X=(F,h) : M \rightarrow {\mathbb{H}^2}\times {\mathbb R}$ be a conformal and minimal immersion.  The map $F:M\rightarrow {\mathbb{H}^2}$ is harmonic and $h$ is a harmonic function on $M$. Let $z$ be a local conformal coordinate on $M$ and let $\sigma$ be the conformal factor of the hyperbolic metric on ${\mathbb{H}^2}$. We set $$Q(F):= (\sigma \circ F)^2 F_z {\overline{F}} _z dz^2 =\phi (z) dz^2,$$ then $Q(F)$ f is a quadratic holomorphic differential globally defined on $M$, known as the [*quadratic Hopf differential associated to $F$*]{}.  Moreover we have $(h_z)^2(dz)^2=-Q(F)$ (see [@ST Proposition 1]), so that $Q(F)$ has two square roots globally defined on $M$. We denote by $\sqrt{\phi}\,dz$ the square root of $Q(F)$, so that $$h=-2\,{{\rm Re}}\int\!\!  i\sqrt{\phi}\,dz =2\, {{\rm Im}}\int\!\!  \sqrt{\phi}\,dz.$$

The metric induced on $M$ by the immersion $X$ is $$ds^2 = (\sigma \circ F)^2 \Big(\rvert F_z \rvert + \rvert F_{{\overline{z}}}
\rvert\Big)^2 \rvert dz\rvert^2.$$

From a result by A. Huber [@Hub Theorem 15], we deduce that a minimal end $E$ of finite total curvature is parabolic, so that it can be parametrized by $U:= \{ z\in {\mathbb C}\mid \ \rvert z \rvert > 1\}$. Let $X=(F,h) : U \rightarrow   {\mathbb{H}^2}\times {\mathbb R}$ be a conformal and complete parametrization of the end, $E=X(U)$.

As it is shown in [@HR], $\phi= (\sigma \circ F)^2 F_z {\overline{F}} _z$ has a finite number of zeroes in $U$, then, up to restrict $U$, we can assume that $\phi$ has no zeros on $U$. Moreover, it is proved in [@HR], that $\phi$ extends meromorphically to the puncture $z=\infty$. Therefore, $\phi$ has the following form

$$\label{Forme generale}
 \phi (z) = \big(\sum_{k \geq 1} \frac{a_{-k}}{z^k} + P(z)\big)^2,$$

where $P$ is a polynomial function.  If we choose $\sqrt{\phi}=\sum_{k \geq 1} \frac{a_{-k}}{z^k} + P(z)$, then $$h=2\, {{\rm Im}}\int \big(\sum_{k \geq 1} \frac{a_{-k}}{z^k} + P(z)\big) dz.$$ Since the height function is well defined on $U$, the real part of $a_{-1}$ is zero. Let $\beta\in{\mathbb R}$ such that $a_{-1}=i\beta.$

\[L.Polynome\] The polynomial function $P$ is not identically zero. 

Assume by contradiction that $P\equiv 0$. If $a_{-1}=0$ we obtain that $$\int_{U } \rvert \phi (z) \rvert dA <\infty,$$ and it is shown in [@HR] that the minimal end $E$ would have finite area.  From [@Frensel] (Theorem 3 and Remark 4) we deduce that for any $p\in E$ and for any real number $\mu < d_E(p,\partial E)$, we have $Area\big(B(p,\mu)\big)
\geq \pi
\mu^2$, where $B(p,\mu)$ is the geodesic disk in $E$ centered at $p$, with radius $\mu$. Considering a suitable diverging sequence of points $(p_n)$ in $E$, we deduce that $E$ has infinite area.  This gives a contradiction. 

Assume now that $a_{-1}\not=0.$ Since $a_{-1}=i\beta,$ we obtain (up to an additive constant) $$h=2\, {{\rm Im}}\int \big(\sum_{k \geq 1} \frac{a_{-k}}{z^k} \big) dz 
= 2\beta   \log \rvert z \rvert + o(1),$$ where $o(1)$ is a function depending of $z$ and $o(1) \to 0$ when $| z| \to \infty$.

For $R>1,$ let $A_R= \{ R \leq \rvert z \rvert \leq 2 R
\}$. Thus, $X(A_R)$ is a  compact and minimal annulus immersed in ${\mathbb{H}^2} \times {\mathbb R}$, whose boundary has two connected components.  For $R$ large enough, the vertical distance between those two boundary components is larger than $2 \pi,$ while the family of the catenoids stays in a slab of height smaller than $\pi$ [@NSWT Proposition 5.1].  Therefore, we can compare $X(A_R)$ with the catenoids and obtain a contradiction by the maximum principle.  This concludes the proof. 

\[D.degre\] Let $m \geq 0$ be the degree of $P$. We will say that [*$E$ is an end of degree $m$*]{}. 

Let $E$ be an end of degree $m.$ Up to a change of variable, we can assume that the coefficient of the leading term of $P$ is one.  Then, for suitable complex number $a_0,\dots, a_{m-1},$ one has

$$\label{Eq.phi}
 P(z)= z^m +a_{m-1} z^{m-1}+ \cdots + a_0 \quad \text{and}\quad 
\sqrt{\phi}=z^m (1 + o(1)).$$

For any $R>1$, we set $U_R:=\{z\in {\mathbb C}\mid \  \rvert z\rvert > R\}$, $S_R:=\{z\in {\mathbb C}\mid \  \rvert z\rvert = R\}=\partial U_R$ and $E_R := X(U_R).$

We set $$W(z):=\int  \sqrt{\phi (z) }\, dz = \int \big(\sum_{k \geq 1}
\frac{a_{-k}}{z^k} + a_0 +
\cdots + z^m\big)\, dz,$$ so that $h(z)=2\,{{\rm Im}}\, W(z)$. If $\beta=0$, the function $W$ is well defined on $U$. If $\beta \not= 0$, the function $W$ is only locally defined and has a real period equal to $-2\pi \beta$. We denote by $\theta \in {\mathbb R}$ a determination of the argument of $z\in U$, therefore $$\label{Eq. Ima}
\frac{1}{2}h(z)= {{\rm Im}}\, W(z) = \beta  \log \rvert z\rvert \,+\, 
\frac{\rvert z\rvert^{m+1}}{m+1}\big( \sin (m+1)\theta + o(1)\big)$$ and, locally $$\label{Eq. Reel}
{{\rm Re}}\,W(z) = -\beta \theta \,+\, 
\frac{\rvert z\rvert^{m+1}}{m+1}\big( \cos (m+1)\theta + o(1)\big).$$

\(1) For any $R\geq 1$, a [*semi-complete curve*]{} in $U_R$ is the image of a map $c: [0,+\infty \mathclose[ \rightarrow U_R$ such that $\rvert c(t)\rvert \xrightarrow[t\to \infty]{} +\infty$.

\(2) Let $c: [0,+\infty \mathclose[ \rightarrow U_R$ be a semi-complete curve and let $\theta_0$ be a real number.  We say that the image of $c$ has the ray $\{ r e^{i\theta_0},\ r>0\}$ as [*asymptotic direction*]{}, if $\theta (t) \xrightarrow[t\to \infty]{}~ \theta_0$, where $\theta(t)$ is the determination of the argument of $c(t)$ in $[\theta_0-\pi,\theta_0+~\pi\mathclose[$.

From formula (\[Eq. Ima\]) above, by a continuity argument we deduce the following facts. 

\[L.Description\] [(1)]{} There exists $R_0>1$ so that, for $k=0,\dots, 2m+~1$ and for any $R\geq R_0$ , the function ${{\rm Im}}\, W$ is strictly monotonous along the pairwise disjoint arcs $$A_k(R):=\big\{ z\in S_R,\   \frac{k\pi}{m+1}
-\frac{\pi}{10(m+1)}< \arg (z)
<\frac{k\pi}{m+1} +\frac{\pi}{10(m+1)}\big\}.$$ [(2)]{} For any fixed $C\in {\mathbb R}$ one has

-   If $(z_n)$ is a sequence of complex numbers such that $\rvert z_n\rvert \to \infty$ and ${{\rm Im}}\,W(z_n)\equiv C$, then $\sin \big((m+1)\arg z_n\big) \to 0$.

-   There exists $r(C) >R_0$ such that, for any $R\geq r(C)$, there are exactly $2m+2$ points $Re^{i\theta_k}$, $k=0,\dots,2m+1$, on the circle $S_R$ verifying ${{\rm Im}}\, W( Re^{i\theta_k})=C$ and $Re^{i\theta_k} \in A_k(R) $.  Moreover, we have $\theta_k \xrightarrow[R\to \infty]{} \frac{k}{m+1}\pi$.

-   For any $R\geq r(C)$, the set $ U_{R} \cap \{{{\rm Im}}\, W(z)=C\}$ is composed of $2m+2$ semi-complete curves $H_k(C,R)$, $k=0,\dots,2m+1$. Moreover $H_k(C,R)$ has the ray $\{re^{i\frac{k}{m+1}\pi},\ r>0\}$ as asymptotic direction. 

Let $k=-1,\dots, 2m+1.$ We take $C=0$ in Lemma \[L.Description\] and define $H_k(R):= H_k(0,R),$ $R_1:=r(0)>R_0.$ Moreover set $\alpha_k:=
\frac{k\pi}{m+1}.$ Then, we deduce the following result. 

\[D.Ouverts modeles\] For any $R\geq R_1$, the level set $ U_{R} \cap \{{{\rm Im}}\, W(z)=0\}$ is composed of the $2m+2$ semi-complete curves $H_k(R),$ $k=0,\dots,2m+1$, having the following properties (see Figure \[F.Hk\]). 

-   Each curve $H_k(R)$ has a unique boundary point that belongs to the open arc $A_k(R)$.

-   Each curve $H_k(R)$ has the ray $\{re^{i\frac{k\pi}{m+1}},\ r>0\}$ as asymptotic direction. 

-   Each curve $H_k(R)$ is contained in the truncated sector $\Delta_k(R)$ defined as follows $$\begin{aligned}
     \Delta_k(R)  &:= \big\{  \rvert z \rvert>R \ \text{and}\ 
     \alpha_k
    -\frac{\pi}{10(m+1)}< \arg (z)
    < \alpha_k  +\frac{\pi}{10(m+1)}\big\}\end{aligned}$$

Let us state some consequences of the properties of the harmonic function ${\rm Im}\,W.$

Let $C_0 >0$ be a real number such that $C_0 > \max \{ \rvert {{\rm Im}}\, W(z) \rvert,\ z\in S_{R_1}\}$. Let $R_2$ be a real number satisfying $R_2 >r(C_0), r(-C_0), R_1$, where $r(C_0), r(-C_0)$ and $R_1=r(0)$ are as in Lemma \[L.Description\].  Note that the set $ U_{R_1} \cap \{{{\rm Im}}\, W(z)=C_0\}$, is composed of $m+1$ proper and complete curves without boundary $L_0^+,\dots,L_{m}^+$ (see Figure \[F.Lk\]). 

For each $j=0,\dots,m$, the level curve $L_j^+$ is contained in the domain of ${\mathbb C}$ which does not contain $0$ and which is bounded by $H_{2j}(R_1)$, $H_{2j+1}(R_1)$ and an arc of $S_{R_1}$ contained in the arc $\{z\in S_{R_1} \mid \alpha_{2j}-\frac{\pi}{10(m+1)} 
< \arg z <   \alpha_{2j+1}+\frac{\pi}{10(m+1)}\}$.

In the same way, the set $ U_{R_1} \cap \{{{\rm Im}}\, W(z)=-C_0\}$ is composed of $m+1$ proper and complete curves without boundary $L_1^-,\dots, L_m^-$. Each level curve $L_j^-$ is contained in the domain of ${\mathbb C}$ which does not contain $0$ and which is bounded by $H_{2j+1}(R_1)$, $H_{2j+2}(R_1)$ and an arc of $S_{R_1}$ contained in the arc $\{z \in S_{R_1} \mid \alpha_{2j+1}-\frac{\pi}{10(m+1)} 
< \arg z <  \alpha_{2j+2}+\frac{\pi}{10(m+1)}\}$, were we set $H_{2m+2}(R_1) := H_0(R_1)$.

For each level curve $L_j^{\pm}$, we denote by $\mathcal{L}_j^\pm$, the connected component of ${\mathbb C}\setminus L_j^{\pm}$ which does not contain the circle $S_{R_1}$.

For each $k=0,\dots,2m+1$ we define the open set $\Omega_k$ setting: $$\label{Eq. Open}
 \Omega_k:= 
\begin{cases}
\mathcal{L}_{\frac{k}{2}-1}^- \cup \mathcal{L}_{\frac{k}{2}}^+ \cup
\Delta_k (R_2)  & \text{if}\ k\ \text{is even},\\
\mathcal{L}_{\frac{k-1}{2}}^+ \cup \mathcal{L}_{\frac{k-1}{2}}^- \cup
\Delta_k (R_2)  & \text{if}\ k\ \text{is odd},\\
\end{cases}$$ where we set $\mathcal{L}_{-1}^-:=\mathcal{L}_m^-$ (see Figure \[Fig3\]). 

By construction, we have that each $\Omega_k$ is a simply connected domain and, setting $\mathcal{U}:= \cup_{k=0}^{2m+1} \Omega_k$, we have $U_{R_2} \subset \mathcal{U} \subset U_{R_1}$. Moreover, we have $$\Omega_k \subset \big\{  \rvert z \rvert>R_1 \ \text{and}\ 
 \alpha_{k-1}
-\frac{\pi}{10(m+1)}< \arg (z)
< \alpha_{k+1}  +\frac{\pi}{10(m+1)}\big\},$$ (recall that $\alpha_{-1}:=-\pi/(m+1)$ and $\alpha_{2m+2}:= 2\pi$). 

We summarize the above construction as follows. 

\[L.Level set\] Let $C$ be a real number, then the following facts hold. 

[(1)]{} If $C > C_0$ then

-   the level set $\{{{\rm Im}}\, W(z)=C\} \cap \mathcal{U}$ is composed of $m+1$ proper and complete curves without boundary $L_0(C),\dots,L_{m}(C)$ satisfying $L_j(C)\subset \mathcal{L}_{j}^+ $ and, therefore, $L_j(C)\subset \Omega_{2j} \cap \Omega _{2j+1}  $, $j=0,\dots, m$.

-   the level set $\{{{\rm Im}}\, W(z)=-C\} \cap \mathcal{U}$ is composed of $m+1$ proper and complete curves without boundary $L_0(-C),\dots,L_{m}(-C)$ satisfying $L_j(-C)\subset \mathcal{L}_{j}^- $ and, therefore, $L_j(-C)\subset \Omega_{2j+1} \cap \Omega _{2j+2}  $, $j=0,\dots, m$, where $\Omega _{2m+2}:=\Omega_0$.

[(2)]{} If $-C_0 \leq C \leq C_0$ then the level set $\{{{\rm Im}}\, W(z)=C\} \cap \mathcal{U}$ is composed of $2m+2$ proper curves $H_k(C)\subset \Omega_k$, $k=0,\dots,2m+1$, satisfying the same properties as the level curves $H_k(R_2)$ in Corollary $\ref{D.Ouverts modeles}$, with $R=R_2$.

\[P.Restriction\] For $k=0,\dots,2m+1$ and for any $z\in \Omega_k$ we choose the determination of $\arg z$ in the interval $\mathopen]\alpha_{k-1} -\frac{\pi}{10(m+1)}, \alpha_{k+1}    +
\frac{\pi}{10(m+1)}\mathclose[$.

Then, the restriction of $W$ to $\Omega_k$ is a well defined complex function, denoted by $W_k$. Furthermore, $W_k : \Omega_k \rightarrow {\mathbb C}$ is one-to-one and defines a conformal diffeomorphism from $\Omega_k$ onto a simply connected domain ${\widetilde}\Omega_k := W_k ( \Omega_k)$ in the $w$ complex plane. 

Since $\Omega_k$ is a simply connected domain which does not contain the origin, the function $W$ is well defined on $\Omega_k$.

Let $z_1, z_2 \in \Omega_k$ be such that $W_k (z_1) = W_k (z_2)$. We deduce from Lemma \[L.Level set\] that for any $C\in {\mathbb R}$, the level set $\{{{\rm Im}}\, W_k (z) =C\}$ has a unique connected component in $\Omega_k$. Therefore, $z_1$ and $z_2$ belong to the same level curve $L\subset \Omega_k$. Since $W_k^\prime (z)=\sqrt{\phi (z)}$ and $\phi$ does not vanish on $\mathcal{U}$, we deduce that the function ${{\rm Re}}\, W$ is strictly monotonous on $L$. We conclude that $z_1=z_2$ as desired. 

In the $w$-complex plane the domains ${\widetilde}\Omega_k, $ $k=0,\dots,2m+1$, defined in Proposition \[P.Restriction\], have a nice structure, that will be crucial in the following. 

\[C.Description\] Let $k$ be an even number, $k=2j$. Then, ${\widetilde}\Omega_k$ is the complementary of a horizontal half-strip.  The non horizontal component of $\partial {\widetilde}\Omega_k$ is a compact arc that is the image by $W_k$ of the boundary arc of $\Omega_k$ in $A_k(R_2)$ joining $L_j^+$ and $L_{j-1}^-$. Thus, ${{\rm Im}}\, W$ is strictly monotonous along such non horizontal component and ${{\rm Re}}\, w$ is bounded from above by a real number $a_k$ for any $w\in \partial  {\widetilde}\Omega_k$ $($see Figure $\ref{F.pair})$.

If $k$ is an odd number, then ${\widetilde}\Omega_k$ has a similar description, except that on the half-strip the real part of $w$ is now bounded from below, i.e. for some real number $b_k$ we have ${{\rm Re}}\, w > b_k$ for any $w\in \partial {\widetilde}\Omega_k$ $($see Figure $\ref{F.impair})$.

We get a proof of Corollary \[C.Description\] by invoking Lemma \[L.Level set\]. 

By the equalities in , we can take $R_2$ in large enough so that $$\label{Eq. Big}
 \frac{1}{2}\rvert z \rvert^m < \sqrt{\rvert \phi (z)\rvert} 
<2\rvert z \rvert^m$$ when $ \rvert z \rvert \geq R_2$. With this choice, we can prove the following result. 

\[L.Comparaison.2\] There is a real constant $c_1>0$ such that, for any $z$ satisfying $ \rvert z \rvert>2R_2$, there exists $k\in \{0,\dots,2m+1\}$ such that $$z\in \Omega_k \quad \text{and} \quad  
d_\phi (z,\partial \Omega_k)>c_1  \,\rvert z \rvert,$$ where $d_\phi$ stands for the distance on $\Omega_k$ with respect to the $\phi$-metric given by $\rvert \phi (z)\rvert \, \rvert dz\rvert^2$.

First assume that $m\geq 1$. Let $z \in \mathcal{U}$ such that $ \rvert z \rvert \geq 2R_2$. Recall that $\alpha_k=\frac{k\pi}{m+1}$ for $k=-1,\dots,2m+2$. There exists a unique $k\in
\{0,\dots,2m+1\}$ such that either $(\alpha_{k}+\alpha_{k+1})/2 \leq \arg z \leq \alpha_{k+1}$ or $\alpha_{k} \leq \arg z \leq (\alpha_{k}+\alpha_{k+1})/2$. Without loss of generality, we can assume that the latter occurs.  Therefore $z\in \Omega_{k}.$

![ The domains ${\widehat}\Omega_k$ for $m=2$ and $k$ even[]{data-label="Fig-chapeau"}](figure8)

For any $k=0,\cdots,2m+1$, we define the following rays: $$\begin{aligned}
 D_k & := \{ \rho e^{i( \alpha_{k+1} -\pi/10(m+1))},\ \rho\geq R_2\}, \\
 D_k^\prime & := \{ \rho e^{i( \alpha_{k-1} +\pi/10(m+1))},\ \rho\geq R_2\}.\end{aligned}$$ By assumption, $z$ belongs to the subdomain ${\widehat}\Omega_k$ of $\Omega_k$ bounded by $D_k$, $D_k^\prime$ and the arc $\Gamma (R_2)$ of $S_{R_2}$ corresponding to $ \alpha_{k-1} +\pi/10(m+1) \leq \theta \leq \alpha_{k+1} -\pi/10(m+1)$ (see Figure \[Fig-chapeau\]). 

Then, we have $$d_\phi (z,\partial \Omega_k) \geq \min \big\{ d_\phi(z,D_k),\
d_\phi(z,D_k^\prime),\ d_\phi(z, \Gamma (R_2))\big\}.$$ Let $\gamma :[0,1] \rightarrow \Omega_k$ be any smooth arc satisfying $\gamma (0)=z$, $\gamma (1) \in D_k$ and $ \rvert \gamma (t)\rvert >R_2$ for any $t\in [0,1]$. Denoting by $L_\phi (\gamma)$ the length of $\gamma$ for the $\phi$-metric and using (\[Eq. Big\]), we have: $$L_\phi (\gamma) =\int_0^1 \sqrt{\rvert \phi (\gamma(t))\rvert}\, 
\rvert \gamma^\prime (t)\rvert\, dt  \geq 
\frac{1}{2} \int_0^1 \rvert \gamma (t)\rvert ^m\rvert \gamma^\prime (t)\rvert\,
dt \geq \frac{R_2^m}{2} \int_0^1 \rvert \gamma^\prime (t)\rvert\, dt,$$ so that $ L_\phi (\gamma) \geq (1/2) R_2^m L(\gamma)$, where $L(\gamma)$ is the Euclidean length of $\gamma$. Since $4\pi/10(m+1) \leq \alpha_{k+1}-\pi/10(m+1) -\arg z \leq 9\pi/10(m+1)<\pi/2$ we get $$L(\gamma) > d(z,D_k) >\sin\left( \frac{4\pi}{10(m+1)}\right)\, 
\rvert z\rvert$$ where $d(z,D_k)$ stands for the Euclidean distance between $z$ and $D_k$. From the last inequality, we deduce $$\label{Eq. Estimate. D}
 d_\phi(z,D_k) \geq  \frac{R_2^m}{2}  \sin\left( \frac{4\pi}{10(m+1)}\right)\, 
\rvert z\rvert.$$

Let now $\gamma :[0,1] \rightarrow \Omega_k$ be any smooth arc satisfying $\gamma (0)=z$, $\gamma (1) \in D_k^\prime$ and $ \rvert \gamma (t)\rvert >R_2$ for any $t\in [0,1]$. In the same way, we can show that $$L_\phi (\gamma) \geq \frac{R_2^m}{2}   L(\gamma) \geq \frac{R_2^m}{2}
d(z,D_k^\prime).$$ Since $9\pi/10(m+1) \leq \arg z -\alpha_{k-1}-\pi/10(m+1)\leq 14\pi/10(m+1)$ we obtain $$\label{Eq. Estimate. D2}
  d_\phi(z,D_k^\prime) \geq \min\big\{ \sin\big( \frac{9\pi}{10(m+1)}\big),
 \sin\big( \frac{14\pi}{10(m+1)}\big)\big\}\, \frac{R_2^m}{2}\, 
\rvert z\rvert.$$ Finally, let $\gamma :[0,1] \rightarrow \Omega_k$ be any smooth arc satisfying $\gamma (0)=z$, $\rvert \gamma (1) \rvert =R_2$ and $ \rvert \gamma (t)\rvert >R_2$ for any $t\in [0,1\mathopen[$. As before we have $$L_\phi (\gamma) \geq \frac{R_2^m}{2} L(\gamma) \geq 
\frac{R_2^m}{2} \big( \rvert z\rvert -R_2\big).$$ Since $\rvert z\rvert   >2R_2$ we get $$\label{Eq. Estimate. S}
 d_\phi(z, S_{R_2}) > \frac{R_2^m}{4}\, \rvert z\rvert.$$ Now, using estimates (\[Eq. Estimate. D\]), (\[Eq. Estimate. D2\]) and (\[Eq. Estimate. S\]), we are done in the case $m\geq 1$.

 

Now we consider the case $m=0$. Then, there are only two domains: $\Omega_0, \Omega_1$, and we have $\alpha_0=0$, $\alpha_1=\pi$ and $\alpha_2=2\pi$.

Let $z \in \mathcal{U}$ such that $ \rvert z \rvert \geq 2R_2$. For some $k\in \{0,1\}$, we have either $(\alpha_{k}+\alpha_{k+1})/2 \leq \arg z \leq \alpha_{k+1}$ or $\alpha_{k} \leq \arg z \leq (\alpha_{k}+\alpha_{k+1})/2$. Without loss of generality, we can assume that the former occurs and that $k=0$, that is: $\pi/2 \leq \arg z \leq \pi$ and, therefore, $z\in \Omega_1$.

We set $$D  := \{ \rho e^{  i\pi/10},\ \rho\geq R_2\}, \qquad
 D^\prime  := \{ \rho e^{ - i\pi/10},\ \rho\geq R_2\}.$$ We have $d(z,D)\geq \rvert z \rvert /2$ and $d(z,D^\prime)\geq d(z,D)$. Moreover, it can be shown in the same way as in the case $m\geq 1$, that $d_\phi(z, S_{R_2}) >  \rvert z \rvert /4$. We obtain that $d_\phi (z,\partial \Omega_k) > \rvert z \rvert /4$, which concludes the proof. 

\[definition-inverse\] For $k=0,\dots,2m+1$, the map $W_k : \Omega_k \rightarrow {\widetilde}\Omega_k$ is a conformal diffeomorphism.  Since $W_k^\prime (z)=\sqrt{\phi (z)}$, $W_k$ is an isometry when $\Omega_k$ is equipped with the $\phi$-metric $\rvert \phi (z)\rvert \, \rvert dz\rvert^2$ and ${\widetilde}\Omega_k$ is equipped with the Euclidean metric $  \rvert dw\rvert^2$.

We denote by $Z_k : {\widetilde}\Omega_k \rightarrow  \Omega_k$ the inverse function of $W_k$.

The image of the level sets of ${\rm Im}\,W$ by the harmonic map $F$

Let $N:=(n_1,n_2,n_3)$ be the unit normal vector field along the end $E$ such that $(X_x,X_y,N)$ has the positive orientation.  We get from [@ST Proposition  4] that $n_3= \frac{\rvert F_z\rvert -\rvert F_{{\overline{z}}}\rvert }
{\rvert F_z\rvert + \rvert F_{{\overline{z}}}\rvert }$.  We define a function (possibly with poles) $\omega$ on $U_R$ setting [@HST Formula  14 ]

$$\label{F.n3}
  n_3= \tanh \omega.$$

For $k=0,\dots,2m+1$, we denote the restriction of $\omega$ to $\Omega_k$ by $\omega_k $.  The function ${\widetilde}\omega_k : {\widetilde}\Omega_k \rightarrow {\mathbb R}$ is defined by setting ${\widetilde}\omega_k (w):= (\omega_k \circ Z_k)(w)$ for any $w\in {\widetilde}\Omega_k$.

The induced metric $ds^2$ on $U$ reads as $$\label{F.metrique}
 ds^2 = 4{\cosh}^2 (\omega) \, \rvert \phi \rvert \, \rvert dz \rvert^2,$$ see [@HST Equation 14]. 

\[analytic-rem\] Since $\phi$ has no zero on $U$, the function $\omega$ has no pole and the tangent plane of $E$ is never horizontal.  This means that the end $E$ is transversal to any slice ${\mathbb H}^2\times \{t\}.$ Thus, the intersection of $E$ with any slice is composed of analytic curves. 

Let us denote by $\Delta_z$ (resp.  $\Delta_w$) the laplacian on $\Omega$ (resp.  ${\widetilde}\Omega$) with respect to the Euclidean metric $\rvert dz \rvert^2$ (resp.  $\rvert dw \rvert^2$).  Since $\Delta_z \omega_k=2{\sinh}(2\omega_k)\, \rvert \phi (z)\rvert$ (see [@HST Equation 13]), we deduce $$\label{Eq. Bochner}
 \Delta_w {\widetilde}\omega_k = 2{\sinh}(2{\widetilde}\omega_k).$$

For any $w\in {\widetilde}\Omega_k$ we denote by $d_k(w)$ the Euclidean distance between $w$ and the boundary of ${\widetilde}\Omega_k$.

 

The following estimate can be found in [@HR] (see also [@Minsky Lemma 3.3]). 

\[P.Estimate\] There exists a constant $K_0>0$ such that for $k=0,\dots,2m+~1$ and for any $w\in {\widetilde}\Omega_k $ with $d_k(w) >1$, we have $$\label{Eq. Estimate}
 \rvert {\widetilde}\omega_k (w) \rvert \leq \frac{K_0}{{\cosh}d_k (w)} < 2K_0 e^{-d_k(w)}.$$ Consequently, the tangent planes to the end become vertical at infinity. 

The last assertion is a consequence of the estimate (\[Eq. Estimate\]) and Lemma \[L.Comparaison.2\]. 

We recall that the energy density of the harmonic function $F$ with respect to the metric $ \rvert \phi (z)\rvert \, \rvert dz\rvert^2  $ on $U$ and the hyperbolic metric on ${\mathbb{H}^2}$ is the real function defined on $U$ by $$e(z):=\frac{(\sigma \circ F)^2(z)}{\rvert \phi (z)\rvert }
\big(\rvert F_z\rvert^2 + \rvert F_{{\overline{z}}}\rvert^2\big)$$ Then one has $$e(z)= \frac{\rvert F_z\rvert}{\rvert F_{{\overline{z}}}\rvert} + 
\frac{\rvert F_{{\overline{z}}}\rvert}{\rvert F_{z}\rvert}=2 {\cosh}2\omega.$$ where the first equality follows from the definition of $\phi$ and the second equality follows from the definition of $\omega$. Observe that $e^{2\omega}= \frac{\rvert F_z\rvert}{\rvert F_{{\overline{z}}}\rvert}$.

For $k=0,\dots,2m+1$ we denote by ${\widetilde}F_k$ the harmonic map ${\widetilde}F_k := F \circ Z_k : {\widetilde}\Omega_k \rightarrow {\mathbb{H}^2}$.

Recall that the relation between the coordinate $z$ in $\Omega$ and the coordinate $w$ in ${\widetilde}\Omega$ is $w=W(z)$ and $\frac{d z}{d w}=1/\sqrt{\phi \circ Z}$. The energy density ${\widetilde}e$ of ${\widetilde}F$ with respect to the Euclidean metric $\rvert dw\rvert^2  $ on ${\widetilde}\Omega$ and the hyperbolic metric on ${\mathbb{H}^2}$ is defined on ${\widetilde}\Omega$ by $${\widetilde}e(w):=(\sigma \circ {\widetilde}F)^2(w)
\big(\rvert {\widetilde}F_w \rvert^2 + \rvert {\widetilde}F_{{\overline{w}}}\rvert^2\big)$$ As before, we have $${\widetilde}e(w)= \frac{\rvert {\widetilde}F_w\rvert}{\rvert {\widetilde}F_{{\overline{w}}}\rvert} + 
\frac{\rvert {\widetilde}F_{{\overline{w}}}\rvert}{\rvert {\widetilde}F_{w}\rvert}=2 {\cosh}2{\widetilde}\omega.$$ Thus $ {\widetilde}e(w)= e(z)$, if $z=Z(w)$.

\[D.courbure\] Let $\kappa : U \rightarrow{\mathbb R}$ be defined as follows: for any $z_0\in U$, $\kappa (z_0)$ is the [*geodesic curvature*]{} in ${\mathbb{H}^2}$ (with respect to the normal orientation induced by the unit normal vector field $N$ on $E$) of the connected component of $F(\{ {{\rm Im}}\, W= {{\rm Im}}\, W(z_0)\})$ passing through the point $F(z_0)$.

For any $k=0,\dots, 2m+1$, let ${\widetilde}\kappa:  {\widetilde}\Omega_k\rightarrow{\mathbb R}$ be defined by setting ${\widetilde}\kappa(w_0)=\kappa(z_0),$ where $w_0=W(z_0)$

As a consequence of Remark \[analytic-rem\], we have that the function $\kappa$ is analytic

Fix a number $k\in\{0,\dots2m+1\}$ and consider the simply connected domain $\Omega_k$ defined in .  Then, setting $w=u+iv$ on ${\widetilde}\Omega_k$, the pullback by the harmonic map ${\widetilde}F_k : {\widetilde}\Omega_k \rightarrow {\mathbb{H}^2}$ of the hyperbolic metric $ \sigma^2 (\xi) \rvert d\xi \rvert ^2$ is given by $$\begin{aligned}
\label{Eq. Metrique}
 {\widetilde}F_k ^* \big(\sigma^2 (\xi) \rvert d\xi \rvert ^2\big) = 4{\cosh}^2 {\widetilde}\omega_k \ du^2 +  4{\sinh}^2 {\widetilde}\omega_k \ dv^2 .\end{aligned}$$

Moreover, for any horizontal coordinate curve ${\widetilde}\gamma:=\{ v=const\}$ in ${\widetilde}\Omega_k$, the absolute value of the geodesic curvature ${\widetilde}\kappa$ of the curve ${\widetilde}F_k({\widetilde}\gamma)$ in ${\mathbb{H}^2}$ is given by $$\label{Eq.geodesic-curvature}
 \rvert {\widetilde}\kappa (w)  \rvert = \frac{1}{2 {\cosh}{\widetilde}\omega_k} 
\left\rvert \frac{\partial {\widetilde}\omega_k}{\partial v}\right\rvert (w)$$ for any $w\in {\widetilde}\gamma$.

A straightforward computation shows that $$F_k^* \big(\sigma^2(\xi) d\xi d{\overline{\xi}}\big)=
\phi (z) dz^2 + {\overline{\phi}} (z) d{\overline{z}} ^2 +e(z)\,\rvert \phi (z)\rvert dz d{\overline{z}} .$$ Since $dw=\sqrt{\phi (z)}dz$ and $d{\overline{w}}= \sqrt{{\overline{\phi}} (z)}d{\overline{z}}$, in the coordinate $w=u+iv$, we have $${\widetilde}F_k^* \big( \sigma^2(\xi) d\xi d{\overline{\xi}}\big)=
({\widetilde}e+2)du^2 + ({\widetilde}e-2)dv^2 
= 4{\cosh}^2 {\widetilde}\omega_k \ du^2 +  4{\sinh}^2 {\widetilde}\omega_k \ dv^2 .$$

Then equality is proved. 

Now, let $w_0\in {\widetilde}\gamma$ and assume ${\widetilde}\omega_k (w_0)\not=0$. Then, by (\[Eq. Metrique\]), the pullback by ${\widetilde}F_k$ of the hyperbolic metric is a regular metric in a neighborhood of $w_0$ in ${\widetilde}\Omega_k$. Consequently, the geodesic curvature of ${\widetilde}F_k({\widetilde}\gamma)$ at $w_0$ is given by $${\widetilde}\kappa (w_0)= -\frac{1}{2} \,\frac{1}{4{\cosh}^2 {\widetilde}\omega_k}
\,\frac{1}{2\rvert {\sinh}{\widetilde}\omega_k \rvert} 
\,\frac{\partial }{\partial v}(4{\cosh}^2 {\widetilde}\omega_k)(w_0)
=-\frac{1}{2} \frac{1}{{\cosh}{\widetilde}\omega_k}
\frac{{\sinh}{\widetilde}\omega_k}{\rvert {\sinh}{\widetilde}\omega_k \rvert}
 \frac{\partial {\widetilde}\omega_k}{\partial v} (w_0),$$ (see [@Kreyszig Formula (42.8)]).  Therefore, the proof is finished in the case ${\widetilde}\omega_k (w_0)\not=0$.

Assume now that ${\widetilde}\omega_k (w_0)=0$. If ${\widetilde}\omega_k$ vanishes identically in a neighborhood of $w_0$, then the tangent plane of the minimal end $E$ is always vertical in a open neighborhood of $X\big(Z_k(w_0)\big)$. This means that such neighborhood is contained in a vertical cylinder in ${\mathbb{H}^2} \times {\mathbb R}$. Since $E$ is minimal, the vertical cylinder is a part of a vertical geodesic plane and, by analyticity, the whole end $E$ is contained in the geodesic plane.  Consequently the curve ${\widetilde}F_k({\widetilde}\gamma)$ is a part of a geodesic of ${\mathbb{H}^2}$ and formula (\[Eq.geodesic-curvature\]) is trivially satisfied. 

If ${\widetilde}\omega_k$ is not identically zero in a neighborhood of $w_0$, then there exists a sequence $(w_n)_{n\in \n^*}$ in ${\widetilde}\Omega_k$ converging to $w_0$ such that ${\widetilde}\omega_k (w_n) \not=0$ for any $n>0$. Since formula (\[Eq.geodesic-curvature\]) holds at any point $w_n$ and $|\kappa|$ is a continuous function, then holds also at $w_0$.

The following Proposition is crucial in order to understand the geometry of the horizontal sections. 

\[P.courbure\] Let $z_0\in U$ and let $\kappa (z_0)$ be the geodesic curvature of the level curve $F(\{{{\rm Im}}\, W(z)={{\rm Im}}\, W(z_0)\})\subset {\mathbb{H}^2}.$ We set $R_3=\max \{2R_2, 2/c_1\}$, where $c_1>0$ is the constant given by Lemma $\ref{L.Comparaison.2}$. Then, there exists a constant $c_2 >0$ such that, for any $z_0\in U_{R_3},$ we have $$\rvert \kappa (z_0) \rvert < c_2 e^{-c_1\rvert z_0\rvert}.$$

Let $z_0 \in U$ be any point such that $\rvert z_0 \rvert > R_3$. It follows from Lemma \[L.Comparaison.2\] that there exists $k\in \{0,\dots,2m+1\}$ such that $$z_0 \in \Omega_k \quad \text{and}\quad 
d_\phi (z_0, \partial \Omega_k)>c_1 \rvert z_0\rvert.$$ Setting $w_0 := W(z_0)\in {\widetilde}\Omega_k$, we get $d_k (w_0):=d_k(w_0, \partial {\widetilde}\Omega_k) = d_\phi (z_0, \partial \Omega_k)$, where $d_k$ denotes the Euclidean distance in ${\widetilde}\Omega_k$. Therefore, we obtain $d_k(w_0) >2,$ since $c_1 \rvert z_0\rvert > c_1 R_3 >2$.

Let ${\widetilde}D$ be the unit disk in the $w$-complex plane, centered at $w_0$, thus ${\widetilde}D \subset {\widetilde}\Omega_k$. For any $w\in {\widetilde}D,$ we denote by $d(w)$ the Euclidean distance between $w$ and $\partial {\widetilde}D$. Recall that the function ${\widetilde}\omega_k$ satisfies Equation (\[Eq. Bochner\]) on ${\widetilde}\Omega_k$. We restrict ${\widetilde}\omega_k$ to ${\widetilde}D$ and we apply the interior [*a-priori*]{} gradient estimate for the Poisson Equation [@G-T Theorem 3.9], then

$$\sup_{{\widetilde}D} \big( d(w) \rvert \nabla {\widetilde}\omega_k\rvert \big)   <
K_1 \big( \sup_{{\widetilde}D} \rvert {\widetilde}\omega_k\rvert + 
2\sup_{{\widetilde}D} d^2 (w) \rvert  {\sinh}2{\widetilde}\omega_k\rvert \big),$$

for some constant $K_1 >0$, where $\nabla$ means the Euclidean gradient. 

Since $d(w_0)=1$ and $d(w)\leq 1$ for any $w\in {\widetilde}D$, we get $$\rvert \nabla {\widetilde}\omega_k\rvert (w_0) < K_1 \big( \sup_{{\widetilde}D} \rvert {\widetilde}\omega_k\rvert + 
2\sup_{{\widetilde}D}  \rvert  {\sinh}2{\widetilde}\omega_k\rvert \big)$$ Moreover, since $d_k( w) \geq d_k (w_0)-1$ for any $w\in {\widetilde}D$, we deduce from Proposition \[P.Estimate\] that $$\rvert {\widetilde}\omega_k (w) \rvert \leq 
\frac{K_0}{{\cosh}(d_k (w_0)-1)}$$ for any $w\in {\widetilde}D$. Using the inequality ${\cosh}(t-1)>\frac{e^t}{10}$ for any $t\in {\mathbb R}$ we obtain $$\rvert \nabla {\widetilde}\omega_k\rvert (w_0) < K_1 \big(
10 K_0 e^{-d_k(w_0)} + 2 {\sinh}( 20K_0 e^{-d_k(w_0)})\big).$$ The function $x\mapsto \frac{{\sinh}x}{x}$ is strictly increasing for $x>0.$ As $d_k(w_0)>2$, then we obtain that ${\sinh}( 20K_0 e^{-d_k(w_0)}) < e^2 {\sinh}(20K_0e^{-2})\, e^{-d_k(w_0)}$. This proves that there exists a constant $\delta >0$ such that $$\label{F.estimee du gradient}
   \rvert \nabla {\widetilde}\omega_k\rvert (w_0) < \delta\, e^{-d_k(w_0)}$$ for any $w_0\in {\widetilde}\Omega_k$ such that $d_k(w_0)>2$. From formula (\[Eq.geodesic-curvature\]) and from the previous computations, setting $c_2:=\delta/2$, we conclude that $$\rvert {\widetilde}\kappa (w_0) \rvert < c_2 e^{-d_k(w_0)}.$$ As $\kappa (z_0)={\widetilde}\kappa (w_0)$ and $d_k(w_0):=d_\phi (z_0,\partial \Omega_k)>c_1 \rvert z_0\rvert$ by Lemma \[L.Comparaison.2\], this completes the proof. 

In view of Lemma \[L.Level set\], let us sum up some notations previously established.  For any $C\in {\mathbb R}$ and for $k=0,\dots,2m+1$, $H_k(C,R)\subset \mathcal{U}\subset U $, denotes the semi-complete level curve of the function $ {{\rm Im}}\, W$ whose asymptotic direction is $\{re^{i\alpha_k},\ r>0\}$, where $\alpha_k=k\pi/(m+1)$. That is $${{\rm Im}}\, W(z)=C\ \text{for any } z\in H_k(C,R) \quad \text{and}\quad
\arg z \xrightarrow[\rvert z\rvert \to \infty ]{}    \alpha_k,  \ \ z\in H_k(C,R).$$

For any $C>C_0$, we denote by $L_j(C)$ (resp.  $L_j(-C)$), $j=0,\dots,m$, the proper and complete level curves given by $\{{{\rm Im}}\, W=C\}$ (resp.  $\{{{\rm Im}}\,
W=-C\}$).  We have, for any $R,$ $H_{2j}(C,R)\cup H_{2j+1} (C,R)\subset L_j(C)$ and $H_{2j+1}(-C,R)\cup H_{2j+2} (-C,R)\subset L_j(-C)$, $j=0,\dots,m$, where $H_{2m+2}(-C,R):=H_0(-C,R)$.

 

The notion of [*convergence in the $C^1$ topology* ]{} in the next Theorem is given in the statement of Definition \[D.C1\]. 

\[T.Description\] [(1)]{} For any $C\in {\mathbb R},$ let $r(C)$ be defined as in Lemma \[L.Description\].  Then, for any $C\in {\mathbb R},$ for any $k\in
\{0,\dots,2m+1\}$ and $R>r(C),$ the level curve $F(H_k(C,R))\subset \mathcal{U} \subset {\mathbb{H}^2}$ is a proper semi-complete curve which has no limit point in ${\mathbb{H}^2}$ and with a unique asymptotic point in ${\partial_{\infty}}{\mathbb{H}^2}$.

[(2)]{} For any $C_1,C_2\in {\mathbb R}$ and for any $k\in \{0,\dots,2m+1\}$, the level curves $F(H_k(C_1,R)),F(H_k(C_2,R)) \subset {\mathbb{H}^2}$ are asymptotic.  That is, there exist a sequence $(p_n)$ on $H_k(C_1,R)$ and a sequence $(q_n)$ on $H_k(C_2,R)$ such that $\rvert p_n \rvert,\  \rvert q_n \rvert \to \infty$, and $d_{{\mathbb{H}^2}}\left( F(p_n), F(q_n)\right) \to 0$. Consequently, $F(H_k(C_1,R))$ and $F(H_k(C_2,R))$ have the same asymptotic point $\theta_k \in~ {\partial_{\infty}}{\mathbb{H}^2}$.

[(3)]{} For $k=0,\dots,2m+1$, the asymptotic points $\theta_k$ and $\theta_{k+1}$ are distinct, $( \theta_{2m+2}:=~\theta_0)$.

[(4)]{} Let $j\in \{0,\dots,m\}$.

-   When $C\to +\infty$, then the proper and complete level curves $F(L_j(C))\subset {\mathbb{H}^2}$ converge for the $C^1$ topology to the geodesic in ${\mathbb{H}^2}$ with asymptotic boundary $\{ \theta_{2j},  \theta_{2j+1}\}$.

-   When $C\to +\infty$, then the proper and complete level curves $F(L_j(-C))\subset {\mathbb{H}^2}$ converge for the $C^1$ topology to the geodesic in ${\mathbb{H}^2}$ with asymptotic boundary $\{ \theta_{2j+1},  \theta_{2j+2}\}$, $(\theta_{2m+2}:=\theta_0)$.

Assertion (1) is a straightforward consequence of the curvature estimates given in Proposition \[P.courbure\], together with Proposition \[P.limit\]. 

Let us prove Assertion (2).  Let $k\in \{0,\dots,2m+1\}$. By Lemma \[L.Level set\], for any $R>r(C_i),$ we have $H_k (C_i,R)\subset \Omega_k$, $i=1,2$. From formula (\[Eq. Reel\]) we deduce that ${{\rm Re}}\, W (z) \xrightarrow[\rvert z\rvert \to \infty]{} 
+ \infty,$ $z\in H_k (C_i,R)$ (resp.  $-\infty$) if $k$ is an even (resp.  odd) number, $i=1,2$.

Assume now that $k$ is even (the argument is analogous in the other case).  Then, by the geometry of the sets $W(\Omega_k)$ (see Corollary \[C.Description\]), setting ${\widetilde}p_n:= n+iC_1$ and ${\widetilde}q_n:=n+iC_2$, for any $n\in \n$ large enough, we have ${\widetilde}p_n \in W\big(H_k(C_1)\big)\subset {\widetilde}\Omega_k$ and ${\widetilde}q_n \in W\big(H_k(C_2)\big) \subset {\widetilde}\Omega_k$. Moreover, setting ${\widetilde}\gamma_n := \{ (1-t){\widetilde}p_n +t {\widetilde}q_n,\ 0\leq t\leq 1\}$, we have ${\widetilde}\gamma_n \subset {\widetilde}\Omega_k$.

Let us set $p_n=Z_k ({\widetilde}p_n)$, $q_n=Z_k ({\widetilde}q_n)$ and $\gamma_n=Z_k ({\widetilde}\gamma_n)$ where $Z_k : {\widetilde}\Omega_k \rightarrow \Omega_k$ is the inverse function of $W$ restricted to $\Omega_k$ [as defined in Remark \[definition-inverse\].]{}  Thus, we have:

-   $\partial \gamma _n =\{p_n,q_n\}$, $p_n\in H_k(C_1,R)$ and $q_n\in H_k(C_2,R)$.

-   $ {{\rm Re}}\, W(z) =n,$ for any $z\in \gamma_n.$

The distance between $ F_k(p_n)$ and $ F_k(q_n)$ in ${\mathbb{H}^2}$ is smaller than the length of $ F_k( \gamma_n)$ which, by construction, is equal to the length of ${\widetilde}F_k({\widetilde}\gamma_n)$.

We need to prove the following Claim. 

Indeed, let $c: \mathopen]0,1\mathclose[\rightarrow \gamma \subset \gamma_n$ be a smooth parametrization of $\gamma$. Since ${{\rm Re}}\, W \big(c(t)\big) \equiv n$ for $0<t<1$, setting $w=W(z)$ and differentiating with respect to $t$, we get $\frac{d w}{dz }\, \frac{d c}{dt} + {\overline{ \left( \frac{d w}{dz }\right)}}\, 
  {\overline{ \left(\frac{d c}{dt}\right)}}=0$. Moreover, as $\omega _k (c(t))\equiv 0$ we have $\rvert F_z\rvert =\rvert F_{{\overline{z}}}\rvert$ along $\gamma$. Recall that $\frac{d w}{dz }=\sqrt{\phi}$ and $\phi = (\sigma \circ F)^2 F_z {\overline{F}} _z$. Combining those relations, we obtain that $F_z\, \frac{d c}{dt} + F_{{\overline{z}}}\, {\overline{ \left(\frac{d c}{dt}\right)}}\equiv 0.$ Then $\frac{d}{dt} (F\circ c)(t) \equiv 0$, which proves the Claim. 

Consequently, the length of $ F_k(\gamma_n)$ in ${\mathbb{H}^2}$ is equal to the length of ${\widetilde}\gamma_n$ with respect to the pseudo-metric (\[Eq. Metrique\]) on ${\widetilde}\Omega_k$, denoted by $L_k ({\widetilde}\gamma_n)$.

Corollary \[C.Description\] yields that for any $w\in {\widetilde}\gamma_n$ and for $n$ large enough we have $$d_k(w):=d_k(w,\partial {\widetilde}\Omega_k) \geq n-a_k,$$ where, as usual, $d_k$ is the Euclidean metric on ${\widetilde}\Omega_k$. We have $$d_{{\mathbb{H}^2}} \big( F_k(p_n), F_k(q_n)\big) = 
d_{{\mathbb{H}^2}} \big({\widetilde}F_k( {\widetilde}p_n),{\widetilde}F_k( {\widetilde}q_n)\big)  \leq 
L_{{\mathbb{H}^2}} \big( {\widetilde}F_k({\widetilde}\gamma_n)\big)= L_k ({\widetilde}\gamma_n)$$

where $d_{{\mathbb{H}^2}}$ (resp.  $L_{{\mathbb{H}^2}}$) is the distance (resp.  the length) in the hyperbolic metric. 

As $u={{\rm Re}}\, W \equiv n$ along ${\widetilde}\gamma_n$, we obtain $$\begin{aligned}
  L_k ({\widetilde}\gamma_n) = 2\rvert C_1 -C_2\rvert \int _0^1 
\rvert {\sinh}{\widetilde}\omega_k \big({\widetilde}\gamma_n (t)\big)\rvert \, dt 
  \leq 2 {\sinh}\left( \frac{K_0}{{\cosh}(n-a_k)}\right) \rvert C_1 -C_2\rvert,\end{aligned}$$ where the inequality comes from formula  (\[Eq. Estimate\]).  Hence, we have $d_{{\mathbb{H}^2}} \big( F_k(p_n), F_k(q_n)\big) \to 0$ when $n\to +\infty$. We conclude that $F(H_k(C_1,R))$ and $F(H_k(C_2,R))$ have the same asymptotic point $\theta_k \in {\partial_{\infty}}{\mathbb{H}^2}$ and this completes the proof of Assertion (2).   

Let us prove Assertion (3).  Assume, for instance, that $\theta_0= \theta_1$. Then, for any $C>C_0$, there exists a complete level curve $L_0(C)\subset \Omega_0$ such that $F(L_0(C)) \subset {\mathbb{H}^2}$ is a proper and complete curve with ${\partial_{\infty}}F(L_0(C))=\{\theta_0\}$. We deduce from Proposition \[P.courbure\] and formula (\[Eq. Ima\]) that for $C$ large enough, [the absolute value]{} of the geodesic curvature of $F(L_0(C)) \subset {\mathbb{H}^2}$ is smaller than 1/4.  Let $\Gamma \subset {\mathbb{H}^2}$ be any complete geodesic such that $\theta_0\in {\partial_{\infty}}\Gamma$. Then, we obtain a contradiction with the maximum principle, comparing $F(L_0(C))$ with the family of complete curves $\gamma_p,\ p\in \Gamma$, orthogonal to $\Gamma$ at $p$, with constant curvature 1/2, and such that $\theta_0$ belongs to the asymptotic boundary of the mean convex component of ${\mathbb{H}^2} \setminus \gamma_p$. This completes the proof of Assertion (3). 

Assertion (4) is a straightforward consequence of Assertion (3), Propositions \[P.courbure\] and \[P.C1\]. 

\(1) We deduce from Theorem \[T.Description\] that the asymptotic boundary of $F(U)$ is composed of exactly $2m+2$ points, counting with multiplicity.  In particular, if $m=0$ then ${\partial_{\infty}}F(U)$ has exactly two distinct points. 

\(2) Observe that the $2m+2$ asymptotic points $\theta_0,\dots,\theta_{2m+1}$ of $F(U)$ need not to be distinct.  They even not need to be well ordered as we can see in some examples found by J. Pyo and M. Rodriguez [@P-R].  We can construct [*artificial*]{} examples for which the asymptotic points are not distinct: just consider the covering maps $\psi_n : U\rightarrow U$, $n \geq
2$, defined by $\psi_n (z)=z^n$, and the minimal ends $X_n:= X\circ \psi_n : U \rightarrow {\mathbb{H}^2} \times {\mathbb R}$.

Let us state two results that have an independent interest in this theory. 

Let $X:= (F,h) : U \rightarrow E\subset {\mathbb{H}^2} \times {\mathbb R}$ be a complete minimal end with finite total curvature.  Then, $E$ is properly immersed. 

Let $m\in \n^*$ be the degree of the end $E$ (see Definition \[D.degre\]).  Let $(p_n)$ be a sequence in $U$ such that $\rvert p_n\rvert \to +\infty$. We want to show that $(X(p_n))$ is not a bounded sequence in ${\mathbb{H}^2} \times {\mathbb R}$. Up to choose a subsequence, we can assume that there is $k\in \{0,\cdots,2m+1\}$ such that, for any $n,$ we have $p_n \in \Omega_k.$ Recall that $h(z) =2{{\rm Im}}\, W(z)$ for any $z$ in $U$.

If $h(p_n)\to \infty$ we are done. 

Assume that the sequence $(h(p_n))$ of real number is bounded.  Thus, up to considering a subsequences, there exists a real number $C_1$ such that $h(p_n)\to C_1$. We set $S(C_1) :=\{ z\in \Omega_k \mid  h(z)=C_1\}$. Thus, $S(C_1)$ is either a complete curve or a semi-complete curve.  As in the proof of Theorem \[T.Description\], we can construct a sequence $(q_n)$ in $\Omega_k$ such that $$\forall n\in \n, \quad q_n \in S(C_1)  \quad \text{and}\quad 
d_{{\mathbb{H}^2}}\big(F(p_n), F(q_n)\big) \to 0.$$ Since $F(S(C_1))$ has no limit point in ${\mathbb{H}^2}$ and has an asymptotic point $p_\infty \in {\partial_{\infty}}{\mathbb{H}^2}$ (see (1) in Theorem \[T.Description\]), we get that $F(q_n) \to p_\infty$ and consequently $F( p_n) \to p_\infty$. Therefore, the sequence $X(p_n)$ is not bounded, which concludes the proof. 

\[T.estimees de courbure\] Let $E\subset {\mathbb{H}^2} \times {\mathbb R}$ be a minimal, complete end with finite total curvature.  Then, there exists a constant $c_3=c_3 (E)$ such that, for any $p\in E$, we have $$\label{F.courbure sur un bout}
 \rvert K_E (p) \rvert \leq c_3 \, e^{-d_E(p,\partial E)},$$ where $K_E$ denotes the intrinsic Gauss curvature and $d_E(\cdot, \partial E)$ stands for the intrinsic distance on $E$.

A straightforward consequence of Theorem \[T.estimees de courbure\] is the following: for any complete and connected minimal surface $\Sigma\subset {\mathbb{H}^2} \times {\mathbb R}$ with finite total curvature, and for any $p_0 \in \Sigma$, there exists a constant $c_4=c_4(p_0,\Sigma)$ such that for any $p\in \Sigma$ we have $$\rvert K_\Sigma (p)\rvert \leq c_4\, e^{-d_\Sigma(p, p_0)}$$

Let $m\in \n$ be the degree of the end $E.$ We consider the open sets $\Omega_k\subset U$, $k=0,\cdots,2m+1$, as defined in and the real number $R_3 >1$ given in Proposition \[P.courbure\].  In this proof we will use the notation previously established for the function $W={{\rm Im}}\, \int \sqrt{\phi (z)}\,
dz,$ ${\widetilde}\Omega_k=W(\Omega_k),$ and $Z_k:{\widetilde}\Omega_k\longrightarrow \Omega_k$.

Let $p\in E$ and let $z\in U$ such that $X(z)=p$. Assume first that $|z|>R_3$. Therefore, by Lemma \[L.Comparaison.2\], there exists $k\in \{0,\dots,2m+1\}$ such that $z\in \Omega_k$ and $d_\phi (z,\partial \Omega_k) >c_1 |z|$. We set $w=W(z)$, so that $w\in {\widetilde}\Omega_k$. We deduce from formula (\[F.metrique\]) that the metric ${\widetilde}ds^2$ induced on ${\widetilde}\Omega_k$ by the minimal immersion ${\widetilde}X := X\circ Z_k  : {\widetilde}\Omega_k \rightarrow {\mathbb{H}^2} \times {\mathbb R}$ is given by $${\widetilde}ds^2 = 4 {\cosh}^2 {\widetilde}\omega_k (w)\,  |dw|^2.$$ Therefore, we obtain $$K_E (p)= K_{{\widetilde}ds^2} (w)= 
-\frac{\tanh {\widetilde}\omega_k}{4{\cosh}^2 {\widetilde}\omega_k}\, \Delta {\widetilde}\omega_k 
- \frac{1}{4{\cosh}^4 {\widetilde}\omega_k}\, |\nabla {\widetilde}\omega_k|^2.$$ It follows from the proof of Proposition \[P.courbure\] that $d_k(w)>2$. Now, by a straightforward computation, using formulas (\[Eq. Bochner\]), (\[Eq. Estimate\]) and (\[F.estimee du gradient\]) (where $\delta=2c_2$), we obtain $$\rvert K_{{\widetilde}ds^2} (w) \rvert  < c_3 e^{-2d_k (w)},$$ for some constant $c_3>0$ which does not depend on $w$. Observe that $$d_E(p,\partial E)= d_E({\widetilde}X (w), \partial E)\geq 
d_E({\widetilde}X (w), {\widetilde}X (\partial {\widetilde}\Omega_k))
=d_{{\widetilde}ds^2} (w,\partial {\widetilde}\Omega_k).$$ From the comparison of the metric $d_{{\widetilde}ds^2}$ with the Euclidean metric, we infer $$d_{{\widetilde}ds^2} (w,\partial {\widetilde}\Omega_k)
\geq 2d_k(w).$$ Formula follows by the previous inequalities for $|z|>R_3$, i.e. outside a compact subset of the end $E$. Finally, it suffices to observe that the continuous function $p\mapsto |K_E (p)| \, e^{d_E(p,\partial E)}$ is bounded on any compact subset of $E$.

Complete minimal surfaces in ${\mathbb{H}^2}\times {\mathbb R}$ with finite total curvature {#Sec. Surface}
===========================================================================================

The aim of this section is to prove the Main Theorem stated in the Introduction.  The proof make essential use of the geometric properties of the horizontal sections of a finite total curvature end, that were established in Section \[Sec. End\]. 

In the following, ${\mathbb{H}^2} \times \{0\}$ will be identify with ${\mathbb{H}^2}.$

Let $X:=(F,h) : U \rightarrow {\mathbb{H}^2} \times {\mathbb R}$ be a conformal and complete minimal annular end, where $U:=\{|z| >1\}$, and let $\gamma \subset {\mathbb{H}^2}$ be a geodesic. 

We say that the end $X(U)\subset   {\mathbb{H}^2}
\times {\mathbb R}$ is [*asymptotic*]{} to the vertical geodesic plane $\gamma \times {\mathbb R}$ if, for any real number $C$ with $|C|$ large enough, $X^{-1}(\{ t=C\})$ is a complete curve of $U$ and if, for any $\varepsilon >0$, there exists a compact subset $K\subset U$ such that the distance between any point of $X(U\setminus K)$ and $\gamma \times {\mathbb R}$ is smaller than $\varepsilon$.

We deduce from Lemma \[L.Polynome\] and Theorem \[T.Description\] that an annular minimal end with finite total curvature is asymptotic to a vertical geodesic plane if, and only if, $m=0.$ That is, if $P$ is a nonzero constant polynomial function. 

\[D.horizontal graph\] Let $\gamma \subset {\mathbb{H}^2}$ be a geodesic.  We say that a nonempty set $S \subset {\mathbb{H}^2} \times {\mathbb R}$ is a [*horizontal graph with respect to the geodesic $\gamma$*]{}, if for any equidistant line ${\widetilde}\gamma$ of $\gamma$ and for any $t \in {\mathbb R}$, the curve ${\widetilde}\gamma \times
\{t\}$ intersects $S$ at most at one point. 

We notice that a different notion of horizontal graph appears in [@SaEarp], in order to treat different kinds of problems about minimal surfaces in ${\mathbb{H}^2}\times{\mathbb R}.$

\[P.non existence\] Let $\gamma_1$ and $\gamma_2$ be two distintc geodesics in ${\mathbb{H}^2}$ with a common asymptotic point.  Then, there is no complete, connected, immersed minimal surface with finite total curvature and two ends, one being asymptotic to $\gamma_1 \times {\mathbb R}$ and the other asymptotic to $\gamma_2 \times {\mathbb R}$.

(see Figure \[Fig4\]).  We set ${\partial_{\infty}}\gamma_1=\{a_\infty, p_\infty\}$ and ${\partial_{\infty}}\gamma_2=\{b_\infty, p_\infty\}$, so that $p_\infty$ is the common asymptotic point of $\gamma_1$ and $\gamma_2$. We denote by $\gamma_\bot$ the geodesic such that $\gamma_1$ is the reflection of $\gamma_2$ across $\gamma_\bot$, we then have $p_\infty \in {\partial_{\infty}}\gamma_\bot$. We denote by $\gamma_0$ the geodesic such that ${\partial_{\infty}}\gamma_0=\{a_\infty, b_\infty\}$. Observe that $\gamma_0$ meets $\gamma_\bot$ orthogonally at some point $p_0\in \gamma_0\cap \gamma_\bot$. For any $s>0$ we denote by $p_s$ the point in the half geodesic $[p_0, p_\infty\mathclose[\subset \gamma_\bot$ such that $d_{{\mathbb{H}^2}}(p_0,p_s)=s$.

For any $s>0$ let $\gamma_s$ be the geodesic orthogonal to $\gamma_\bot$ at $p_s$. We set $P_s := \gamma_s \times {\mathbb R}$.

![[]{ data-label="Fig4"}](figure5)

Assume by contradiction that there exists a complete and connected minimal surface $\Sigma$ with finite total curvature and two ends, one asymptotic to $\gamma_1 \times {\mathbb R}$ and the other asymptotic to $\gamma_2 \times {\mathbb R}$. By a result from A. Huber [@Hub Theorems 13 and 15], such surface is parametrized by a Riemann surface $M$ conformally equivalent to a compact Riemann surface ${\overline{M}}$ punctured at two points $z_1$, $z_2$, $M \simeq {\overline{M}} \setminus \{z_1,z_2\}$. We denote by $X=(F,h) : M \rightarrow \Sigma\subset {\mathbb{H}^2} \times {\mathbb R}$ the minimal and conformal immersion.  Thus, $F : M \rightarrow {\mathbb{H}^2}$ is a harmonic map and $h : M \rightarrow {\mathbb R}$ is a harmonic function. 

We want to show that $\Sigma$ is a horizontal graph with respect to $\gamma_\bot$, afterwards we will derive a contradiction to conclude that such surface does not exist. 

For any $s>0$, we denote by $P_s^-$ the component of $({\mathbb{H}^2}\times {\mathbb R}) \setminus P_s$ containing $\{p_0\} \times {\mathbb R}$ and we denote by $P_s^+$ the other component.  Thus $\{p_\infty\}\times {\mathbb R}\subset {\partial_{\infty}}P_s^+$. For any $s>0$ we set $\Sigma_s^- := \Sigma \cap P_s^-$, $\Sigma_s^+ := \Sigma \cap P_s^+$ and we denote by $\Sigma_s^{-*}$ the reflection of $\Sigma_s^-$ across the vertical geodesic plane $P_s$.

For any $\rho >0$, we denote by $L_\rho^1$ (resp.  $L_\rho^2$) the equidistant line of $\gamma_\bot$ with distance $\rho$, intersecting $\gamma_1$ (resp.  $\gamma_2$).  We denote by $\mathcal{C}_\rho$ the domain of ${\mathbb{H}^2} \times {\mathbb R}$ bounded by $ (L_\rho^1 \cup L_\rho^2 )\times {\mathbb R}$ and we set $\mathcal{Q}_\rho := ({\mathbb{H}^2} \times {\mathbb R}) \setminus \mathcal{C}_\rho$. Thus, $\{a_\infty\}\times{\mathbb R}, \{b_\infty\}\times{\mathbb R}\subset {\partial_{\infty}}\mathcal{Q}_\rho  $ for any $\rho >0$. Since each end of $\Sigma$ is asymptotic to one of the vertical planes $\gamma_i \times {\mathbb R}$, $i=1,2$, for any $s>0$ there exists $\rho_s>0,$ large enough, such that $\Sigma \cap \mathcal{Q}_{\rho_s} \subset P_s^-$.

Therefore, as a consequence of Propositions \[P.courbure\] and \[P.graph\], we have that if $s>0$ is small enough, then for any $t\in {\mathbb R}$, the level set $\Sigma_{s}^- \cap ({\mathbb{H}^2} \times\{t\})$ is a horizontal graph with respect to the geodesic $\gamma_\bot$. Consequently, there exists $s_1>0$ such that $\Sigma_{s_1}^-$ is a horizontal graph with respect to $\gamma_\bot$ and $ \Sigma_s^{-*} \cap \Sigma_s^+ =\emptyset  $ for any $0< s < s_1$.

We set $$I:=\{ \sigma \geq  0 \mid \Sigma_s^{-*} \cap \Sigma_s^+ =\emptyset,\ 
\text{for any } 0<s \leq \sigma \}.$$

In order to ensure that $\Sigma$ is a horizontal graph, we must show that $I=[0,+\infty\mathclose[.$

The set $I$ is nonempty because $[0,s_1]\subset I.$

We set $s_2 =\sup I$. If $s_2=+\infty$ we are done.  Assume that $s_2\not=+\infty$. By a continuity argument we have $\Sigma_{s_2}^{-*} \cap \Sigma_{s_2}^+ =\emptyset$, so that $s_2 \in I$.

Recall that one end of $\Sigma$ is asymptotic to $\gamma_1 \times {\mathbb R}$ and the other end is asymptotic to $\gamma_2 \times {\mathbb R}$. Moreover, from Lemma \[L.Level set\], formula (\[Eq. Ima\]) and Proposition \[P.courbure\], it follows that, for any $\varepsilon >0$, there exists $t_0 >0$ such that for any $t>t_0$, the intersection $\Sigma \cap ({\mathbb{H}^2} \times \{t\})$ is composed of two complete curves, $c^t_1$ and $c^t_2$ verifying $${\partial_{\infty}}c^t_1 =\{a_\infty, p_\infty\},\  {\partial_{\infty}}c^t_2 =\{b_\infty,
p_\infty\}\quad  \text{and}\quad \sup_{c^t_i}\rvert \kappa (q)\rvert <
\varepsilon,\ 
i=1,2,$$ where $\kappa$ denotes the geodesic curvature.  From Proposition \[P.C1\] we deduce that that $c^t_i$ is $C^1$-close to $\gamma_i$, $i=1,2$, if $\varepsilon$ is small enough.  Analogously $\Sigma \cap ({\mathbb{H}^2}
\times
\{-t\})$ is composed of two complete curves, $c^{-t}_1$ and $c^{-t}_2$, $C^1$-close to $\gamma_1$ and $\gamma_2$ respectively. 

We set $p_1^t=c_1^t \cap P_{s_2}$. Observe that the (nonoriented) angle between $c_1^t$ and the equidistant line to $\gamma_\bot$ passing through $p_1^t$ is close to the angle between the same equidistant line and $\gamma_1$ at the common point.  Hence, there is $\alpha\in(0,\pi/2)$ such that, if $t_0$ is large enough, this angle is larger than $\alpha$ for any $t>t_0$, and the same is true for the analogous angle defined for $P_{s_2}\cap c_2^t$, $P_{s_2}\cap c_1^{-t}$ and $P_{s_2}\cap c_2^{-t}$. Therefore, there exists $\eta_1 >0$ such that $\Sigma_{s_2+ \eta_1}^- \cap ({\mathbb{H}^2} \times \{\rvert t\rvert > t_0\})$ is a horizontal graph with respect to $\gamma_\bot$ and $\big( \Sigma_{s_2+ \eta_1}^{-*} \cap \Sigma_{s_2+ \eta_1}^+\big) \cap 
 ({\mathbb{H}^2} \times \{\rvert t\rvert > t_0\})= \emptyset.$ Then the claim is proved. 

Observe first that, at any point of $\Sigma\, \cap P_{s_2}$, the equidistant line to $\gamma_\bot$ passing through this point is not tangent to $\Sigma$. Indeed, suppose that at some point $p\in \Sigma \cap  P_{s_2}$ the equidistant line to $\gamma_\bot$ passing through $p$ is tangent to $\Sigma$. Thus $\Sigma$ is orthogonal to $P_{s_2}$ at $p$ and, therefore, $\Sigma_{s_2}^{-*} $ and $\Sigma_{s_2}^+$ are tangent at the point $p$ of their common boundary.  Since $\Sigma_{s_2}^{-*} \cap \Sigma_{s_2}^+ =\emptyset$, the boundary maximum principle would imply that $\Sigma_{s_2}^{-*} = \Sigma_{s_2}^+$. This gives a contradiction, since the asymptotic boundary of $\Sigma$ is not symmetric with respect to any vertical geodesic plane $P_s$.

Therefore, since $(\Sigma \cap  P_{s_2}) \cap \{   \rvert t \rvert
\leq t_0\}$ is compact, there is $\beta\in(0,\pi/2)$ such that the nonoriented angle between the equidistant lines to $\gamma_\bot$ and $\Sigma$ at any point of $(\Sigma \cap P_{s_2}) \cap \{  \rvert t \rvert \leq t_0\}$ is larger than $\beta$. By a compactness argument again, there is $\eta_2>0$ such that $\Sigma_{s_2+ \eta_2}^- \cap ({\mathbb{H}^2} \times \{\rvert t\rvert \leq  t_0\})$ is a horizontal graph and $\big( \Sigma_{s_2+ \eta_2}^{-*} \cap \Sigma_{s_2+ \eta_2}^+\big) \cap 
 ({\mathbb{H}^2} \times \{\rvert t\rvert \leq t_0\})= \emptyset.$ This proves the claim. 

We set $\eta =\min\{\eta_1,\eta_2\}$. From Claims 1 and 2, we get that $s_2+\eta \in
I$. This gives a contradiction with the maximality of $s_2$. Therefore, $I=[0,+\infty\mathclose[$ and $\Sigma$ is a horizontal graph with respect to $\gamma_\bot$.

Now we can conclude the proof. 

Let $Q(F)$ be the quadratic Hopf differential associated to $F$. We know that $Q(F)$ is holomorphic on $M$ and has a pole at the ends $z_1, z_2 \in {\overline{M}}$. Since each end is asymptotic to a vertical geodesic plane, the degree of each end is $m=0$. Therefore, each end is a pole of order 4 of $Q(F)$. According to the Riemann relation for $Q(F),$ we have that

$$Pole(Q(F)) -Zero (Q(F))=2\chi ({\overline{M}}),$$

thus $ Zero (Q(F)) =Pole(Q(F))-2\chi ({\overline{M}})=8-2\chi ({\overline{M}})\geq 4$. Consequently, there exists $z_0\in M$ which is a zero of $Q(F)$. From formula (\[F.metrique\]) we deduce that $z_0$ is a pole of $\omega$ and then, the tangent plane of $\Sigma$ at $X(z_0)$ is horizontal (see formula ). 

Let $s^\prime >0$ such that $X(z_0)\in P_{s^\prime}$. We get a contradiction by the boundary maximum principle since, on one hand $ \Sigma_{s^\prime}^{-*} \cap \Sigma_{s^\prime}^+ = \emptyset$ (since $\Sigma$ is a horizontal graph with respect to $\gamma_\bot$) but on the other hand $ \Sigma_{s^\prime}^{-*}$ and $\Sigma_{s^\prime}^+$ are tangent at their common boundary point $X(z_0)$.

\[P.non existence 2\] Let $\gamma_1$ and $\gamma_2$ be two distinct geodesics in ${\mathbb{H}^2}$ intersecting at some point.  Then, there is no complete, connected, immersed minimal surface with finite total curvature and two ends, one being asymptotic to $\gamma_1 \times {\mathbb R}$ and the other asymptotic to $\gamma_2 \times {\mathbb R}$.

We set $\{w\}:=\gamma_1 \cap \gamma_2$. We denote by $\alpha$ and $\beta$ the two geodesics passing through $w$ such that the reflection of $\gamma_1$ across $\alpha$ is $\gamma_2$ and the reflection of $\gamma_1$ across $\beta$ is $\gamma_2$. Thus, $\alpha$ intersects $\beta$ orthogonally at $w$ (see Figure \[Fig. Prop3.2\]). 

![ $w=\gamma_1\cap\gamma_2=0.$[]{data-label="Fig. Prop3.2"}](figure9)

We choose an orientation on $\alpha$ and $\beta$. For any $s\in {\mathbb R},$ we denote by $p_s$ (resp.  $q_s$) the point of $\alpha$ (resp.  $\beta$) whose signed distance to $w$ is $s$, observe that $q_0=p_0=w$. Furthermore, for any $s\in {\mathbb R},$ we denote by $P_s$ (resp.  $Q_s$) the vertical geodesic plane passing through $p_s$ (resp.  $q_s$) and orthogonal to the geodesic $\alpha$ (resp.  $\beta$), note that $P_0=\beta \times {\mathbb R}$ and $Q_0=\alpha \times {\mathbb R}$.

We set $ {P_0^+}:= \cup_{s> 0}\, P_s$, $ {P_0^-}:= \cup_{s>0}\, P_s$, $ {Q_0^+}:= \cup_{s> 0}\, Q_s$ and ${Q_0^-}:= \cup_{s> 0}\, Q_s$.

Assume that there exists a complete minimal surface $\Sigma$ with finite total curvature and two ends, one being asymptotic to $\gamma_1 \times {\mathbb R}$ and the other being asymptotic to $\gamma_2 \times {\mathbb R}$.

Using the Alexandrov reflection principle with respect to the vertical planes $P_s$, $s\in {\mathbb R}$, we can show, as in the proof of Proposition \[P.non existence\], that $\Sigma \cap {\overline{P_0^+}}$ is a horizontal graph with respect to the geodesic $\alpha$ and so is $\Sigma \cap {\overline{P_0^-}}$. In the same way, we can show that $\Sigma \cap {\overline{Q_0^+}}$ and $\Sigma \cap {\overline{Q_0^-}}$ are both horizontal graphs with respect to the geodesic $\beta$.

Now we proceed as in the proof of [@Schoen Theorem 3, Case 1]. 

We set $L:= P_0 \cap Q_0=\{w\} \times {\mathbb R}$. Since $\Sigma \cap {\overline{P_0^+}}$ and $\Sigma \cap {\overline{P_0^-}}$ are horizontal graphs with respect to $\alpha$, the self intersection set $S$ of $\Sigma$ is contained in $P_0$. By the same argument, we have $S\subset Q_0$, so that $S\subset L$. Since an end of $\Sigma$ is asymptotic to $\gamma_1 \times {\mathbb R}$ and the other end is asymptotic to $\gamma_2 \times {\mathbb R}$, we have $S\not= \emptyset$. By the analyticity of the sets $\Sigma \cap P_0$ and $\Sigma \cap Q_0$, we get that $S=L$. Moreover, since $\Sigma \cap {\overline{P_0^+}}$ and $\Sigma \cap {\overline{P_0^-}}$ are horizontal graphs with respect to $\alpha$, we deduce that $\Sigma \cap Q_0 =L$. Analogously, $\Sigma \cap P_0 =L$. Therefore, $\Sigma \setminus L$ consists of four connected components $\Sigma_i,\ i=1,\cdots,4$ with: $$\Sigma_1 \subset P_0^+ \cap Q_0^+ ,\qquad \Sigma_2 \subset P_0^+ \cap Q_0^-,
\qquad  \Sigma_3 \subset P_0^- \cap Q_0^- \quad \text{and} \quad 
\Sigma_4 \subset P_0^- \cap Q_0^+,$$ Denoting by $\sigma$ the rotation about the vertical geodesic $L$ with angle $\pi$, the reflection principle shows that $\sigma (\Sigma_1)=\Sigma_3$, so that $\Sigma^\prime := \Sigma_1 \cup \Sigma_3$ is a smooth and complete minimal surface embedded in ${\mathbb{H}^2} \times {\mathbb R}$. For the same reason, $\Sigma^{\prime \prime} := \Sigma_2 \cup \Sigma_4$ is also a smooth and complete minimal surface embedded in ${\mathbb{H}^2} \times {\mathbb R}$. Therefore, we get that $\Sigma =\Sigma^\prime \cup \Sigma^{\prime \prime}$ and consequently, the minimal surface is not connected.  We arrive to a contradiction, which concludes the proof. 

Now we can restate the Main Theorem, announced in the Introduction, in the following way. 

\[T.main theorem\] Let $\Sigma$ be a complete, connected minimal surface immersed in ${\mathbb{H}^2} \times {\mathbb R}$ with finite total curvature and two ends.  Assume that each end is asymptotic to a vertical geodesic plane $\gamma_i \times {\mathbb R}$, where each $\gamma_i$, $i=1,2$, is a geodesic. 

Then, we have $ \gamma_1 \cap \gamma_2 =\emptyset$, ${\partial_{\infty}}\gamma_1 \cap {\partial_{\infty}}\gamma_2 =\emptyset$. Furthermore, $\Sigma$ is a properly embedded annulus and is a two ends model surface. 

In order to prove Theorem \[T.main theorem\] we fix some notations and prove some lemmas. 

Let $\gamma_1, \gamma_2 \subset {\mathbb{H}^2}$ be two geodesics satisfying $$\label{geodesiques disjointes}
  \gamma_1 \cap \gamma_2 =\emptyset \quad \textrm{and} \quad 
{\partial_{\infty}}\gamma_1 \cap {\partial_{\infty}}\gamma_2 =\emptyset.$$ We denote by $\gamma_0 \subset {\mathbb{H}^2}$ the geodesic orthogonal to both $\gamma_1$ and $\gamma_2$. We set $p_1=\gamma_1 \cap \gamma_0$ and $p_2=\gamma_2 \cap \gamma_0$. We call $p_0$ the middle point of the segment of $ \gamma_0$ between $p_1$ and $p_2.$ We denote by $\Gamma\subset {\mathbb{H}^2}$ the geodesic passing through $p_0$ and orthogonal to $\gamma_0$ (see Figure \[Fig. Notations\]). 

![[]{ data-label="Fig. Notations"}](figure6)

In the following lemmas the surface $\Sigma$ satisfies the hypothesis of Theorem \[T.main theorem\]. 

\[L.symetrie-1\] Suppose that $\gamma_1$ and $\gamma_2 $ are geodesics satisfying the properties in $(\ref{geodesiques disjointes})$. Then, the surface $\Sigma$ is symmetric with respect to the vertical geodesic plane $\gamma_0 \times {\mathbb R}$ and each component of $\Sigma\setminus (\gamma_0 \times {\mathbb R}) $ is a horizontal graph with respect to $\Gamma$.

We choose an orientation on $\Gamma$. For any $s\in {\mathbb R}$, we denote by $q_s$ the unique point in $\Gamma$ whose signed distance to $p_0$ is $s$, thus $q_0=p_0$. For any $s\in {\mathbb R},$ we denote by ${\widetilde}\gamma_s \subset {\mathbb{H}^2}$ the geodesic orthogonal to $\Gamma$ and passing through $q_s.$ Observe that ${\widetilde}\gamma_0=\gamma_0$.

For any $s\in {\mathbb R},$ we set $Q_s:= {\widetilde}\gamma_s \times {\mathbb R}$. Moreover, for any $s\not=0,$ we denote by $Q_s^+$ the component of $({\mathbb{H}^2}\times {\mathbb R}) \setminus Q_s$ containing $\{p_0\} \times {\mathbb R}$ and by $Q_s^-$ the other component.  For any $s\not=0$ we set $\Sigma_s^- := \Sigma \cap Q_s^-$, $\Sigma_s^+ := \Sigma \cap Q_s^+$ and we denote by $\Sigma_s^{-*}$ the reflection of $\Sigma_s^-$ across the vertical geodesic plane $Q_s$.

As in the proof of Proposition \[P.non existence\] (Claim 1 and Claim 2), we can show that for any $s \not=0$, $\Sigma_s^-$ is a horizontal graph with respect to $\Gamma$ and that $\Sigma_s^{-*}\cap\Sigma_s^+ =\emptyset$.

Then, passing to the limit for $s\to 0$ from both sides, we conclude that $\Sigma$ is symmetric with respect to $\gamma_0 \times {\mathbb R}$ and that each component of $\Sigma\setminus (\gamma_0
\times {\mathbb R}) $ is a horizontal graph with respect to $\Gamma.$

It follows from the proof of Lemma \[L.symetrie-1\] that the tangent plane at any point of $\Sigma\setminus (\gamma_0 \times {\mathbb R}) $ is never horizontal. 

\[L.symetrie-2\] Suppose that $\gamma_1$ and $\gamma_2 $ are geodesics satisfying the properties in $(\ref{geodesiques disjointes})$. Then, the surface $\Sigma$ is symmetric with respect to the vertical geodesic plane $\Gamma \times {\mathbb R}$ and each component of $\Sigma\setminus (\Gamma \times {\mathbb R}) $ is a horizontal graph with respect to $\gamma_0$.

Let $d>0$ be the distance between $p_0$ and $p_1$, thus $d=d_{{\mathbb{H}^2}} (p_0,p_1)=d_{{\mathbb{H}^2}} (p_0,p_2)$. For any $s \in [0,d]$ we denote by ${\widetilde}p_s \in \gamma_0$ the unique point between $p_0$ and $p_1,$ whose distance to $p_0$ is $s.$ Thus ${\widetilde}p_0=p_0$ and ${\widetilde}p_d=p_1$.

We denote by $\Gamma_s\subset {\mathbb{H}^2}$ the geodesic orthogonal to $\gamma_0$ and passing through ${\widetilde}p_s$, thus $\Gamma_d=\gamma_1$. We set $P_s:= \Gamma_s \times {\mathbb R}$. For any $s \in [0,d \mathclose[$ we denote by $P_s^-$ the connected component of $({\mathbb{H}^2} \times {\mathbb R})\setminus P_s$ containing $\{p_1\} \times {\mathbb R}$ and by $P_s^+$ the other component.  We set $\Sigma_s^-:= \Sigma \cap P_s^-$ and $\Sigma_s^+:= \Sigma \cap P_s^+$. Furthermore, $\Sigma_s^{-*}$ denotes the reflection of $\Sigma_s^-$ across $P_s$. We give to the geodesic $\gamma_0$ the orientation going from $p_1$ to $p_2$.

We will say that $\Sigma_s^{-*} \leq\Sigma_s^+$ if $\Sigma_s^{-*}$ remains under $\Sigma_s^+$ with respect to the orientation of $\gamma_0$. Using Propositions \[P.courbure\] and \[P.graph\], it can be show that there exists $\varepsilon >0$ such that for any $s\in [d-\varepsilon, d\mathopen[\,$, $\Sigma_s^-$ is a horizontal graph with respect to $\gamma_0$. Therefore, for any $s\in [d-\varepsilon/2, d\mathopen[$, we have $\Sigma_s^{-*}
\leq\Sigma_s^+$.

We set $$I=\big\{ s\in [0,d] \mid \Sigma_r^{-*} \leq\Sigma_r^+ \ \text{for any}\ 
r \in \mathclose]d-s, d\mathopen[ \big\}.$$ We have $I\not= \emptyset$, since $\varepsilon/2 \in I$. We set $s_0:= \sup I$ and want to prove that $s_0=d$.

Assume that $s_0\not=d$. By continuity we get that $\Sigma_{s_0}^{-*} \leq\Sigma_{s_0}^+.$ On the other hand we have $\Sigma_{s_0}^{-*} \not= \Sigma_{s_0}^+,$ since the asymptotic boundaries of those two parts are not equal.  Observe that $\partial\Sigma_{s_0}^{-*} = \Sigma \cap
P_{s_0}$ is compact and the boundary maximum principle shows that $\Sigma$ is never orthogonal to $ P_{s_0}$ along their intersection.  We deduce that there exists $\varepsilon_1 >0$ such that $\Sigma_{d-s_0-\varepsilon_1  }^-$ is a horizontal graph with respect to $\gamma_0$ and $    \Sigma_r^{-*} \leq\Sigma_r^+ $ for any $r \in \mathclose]d-s_0-\varepsilon_1, d\mathopen[  $, which gives a contradiction with the maximality of $s_0$. We deduce that $s_0=d$ and then $ \Sigma_0^{-*}
\leq\Sigma_0^+ $. 

Using the same arguments coming from the other side, that is from $p_2$ to $p_0$, we can show that $\Sigma_0^{+*} \geq\Sigma_0^- $.  We conclude that $\Sigma_0^{+*}
=\Sigma_0^-$, that is $\Sigma$ is symmetric with respect to $P_0=\Gamma\times {\mathbb R}$, as desired. 

\[L.symetrie-3\] Suppose that $\gamma_1$ and $\gamma_2 $ are geodesic satisfying the properties in $(\ref{geodesiques disjointes})$. Then, the surface $\Sigma$ is symmetric with respect to some slice ${\mathbb{H}^2} \times \{t_0\}$ and each component of $\Sigma \setminus ({\mathbb{H}^2} \times \{t_0\})$ is a vertical graph. 

From Lemma \[L.symetrie-1\] and \[L.symetrie-2\] we know that each component of $\Sigma\setminus (\Gamma\times {\mathbb R})$ is a horizontal graph with respect to $\gamma_0$ and each component of $\Sigma\setminus (\gamma_0\times {\mathbb R})$ is a horizontal graph with respect to $\Gamma$. Consequently, the intersection $\mathcal{C}:=\Sigma \cap (\Gamma\times {\mathbb R})$ is composed of a finite number of Jordan curves whose interiors are mutually disjoint. 

For $i=1,2$, we call $\Sigma_i$ the component of $\Sigma\setminus (\Gamma\times {\mathbb R})$ which is asymptotic to the vertical plane $\gamma_i \times {\mathbb R}$. For any $t\in {\mathbb R}$, we set $\Pi_t := {\mathbb{H}^2} \times \{t\}$. Let $t_1\in {\mathbb R}$ be such that $\partial\Sigma_1 \cap \Pi_t =\emptyset$ for any $t >t_1$ and $\partial\Sigma_1 \cap \Pi_{t_1 }\not=\emptyset$. Such $t_1$ exists since $\partial\Sigma_1= \mathcal{C}$ is compact. 

For any $t\in {\mathbb R}$ we set: $\Pi_t^+:= {\mathbb{H}^2} \times \{s \mid s>t \}$, $\Pi_t^-:= {\mathbb{H}^2} \times \{s \mid s< t \}$, $\Sigma_{1,t}^+=\Sigma_1 \cap \Pi_t^+ $, $\Sigma_{1,t}^-=\Sigma_1 \cap \Pi_t^- $ and we denote by $\Sigma_{1,t}^{+*}$ the reflection of $\Sigma_{1,t}^+$ across $\Pi_t$. Moreover, $\Sigma_{1,t}^{+*} \geq \Sigma_{1,t}^-  $ means that $\Sigma_{1,t}^{+*}$ stays above $\Sigma_{1,t}^- $. 

Indeed, for $t>t_1$ we know from Lemma \[L.symetrie-2\] that the intersection $\Sigma_1 \cap \Pi_t$ is a complete curve, that is a horizontal graph with respect to $\gamma_0$ and whose asymptotic boundary is ${\partial_{\infty}}\gamma_1 \times \{t\}$. For any $s\in {\mathbb R}$ we denote by $T_s$ the horizontal translation along $\gamma_0$ of signed length $s$ in the direction going from $p_2$ to $p_1$.

Suppose that $\Sigma_{1,t}^{+*}$ does not remain above $\Sigma_{1,t}^-  $.  Then, for $\varepsilon >0$ small enough, the translated $T_\varepsilon
(\Sigma_{1,t}^{+*})$ does not remain above $\Sigma_{1,t}^-  $.  Observe that ${\partial_{\infty}}T_\varepsilon (\Sigma_{1,t}^{+*}) \cap 
{\partial_{\infty}}\Sigma_{1,t}^-=~\emptyset$. Moreover, $\partial T_s (\Sigma_{1,t}^{+*}) \cap {\overline{   \Sigma_{1,t}^- }}=\emptyset$, for any $s>0$, since $\Sigma_1$ is a horizontal graph with respect to $\gamma_0$. We deduce that, for $\varepsilon >0$ small enough, the part of $\Sigma_{1,t}^-  $ which remains above $T_\varepsilon (\Sigma_{1,t}^{+*})$ has compact closure.  Therefore, there exists $s_1>0$ such that $ T_s (\Sigma_{1,t}^{+*}) \cap  \Sigma_{1,t}^-  =\emptyset$ for any $s>s_1$ and $ T_{s_1} (\Sigma_{1,t}^{+*}) \cap  \Sigma_{1,t}^-  \not=\emptyset$. This means that $ T_{s_1} (\Sigma_{1,t}^{+*})$ and $ \Sigma_{1,t}^- $ are tangent at some point and one surface remains in one side of the other, which gives a contradiction with the maximum principle and proves the claim. 

For any $ s\in {\mathbb R}$, let $T_s$ be the horizontal translation defined as in Claim 1.  Since $\Sigma_1$ is a horizontal graph with respect to $\gamma_0$, we have $T_s \left( \mathcal{C}\right) \cap \Sigma_1=\emptyset $ for any $s>0$. Furthermore, since the whole surface $\Sigma$ is symmetric with respect to $\Gamma \times {\mathbb R}$ we have that $T_s \left( \mathcal{C}\right) \cap \Sigma_1=\emptyset $ for any $s\not=0$. Let $\mathcal{D}\subset \Gamma \times {\mathbb R}$ be the bounded domain with boundary $\mathcal{C}$. Since $\Sigma$ is connected, we have $T_s (\mathcal{D}) \cap \Sigma=\emptyset$ for any $s\not=0$.

Let $\varepsilon>0$ such that $\mathcal{C}_{t_1 -\varepsilon}^{+*} \geq \mathcal{C}_{t_1-\varepsilon}^-$, where $\mathcal{C}_{t_1 -\varepsilon}^{+,*}$ etc.  are obviously defined.  Then, using an argument analogous to that of Claim 1, considering translations $T_s$, it can be shown that $\Sigma_{1,t_1 -\varepsilon}^{+*} \geq \Sigma_{1,t_1-\varepsilon}^-  $.  This proves Claim 2. 

Now we can conclude the proof. 

Since $\mathcal{C}$ is composed of a finite number of Jordan curves, there exists a component, say $C$, and a real number $t_0 <t_1$ satisfying $\Sigma_{1,t}^{+*} \geq \Sigma_{1,t}^-  $ for any $t>t_0$, such that at least one of the following properties occurs:

1.   $C_{t_0}^{+*} \geq C_{t_0}^-  $ and $C_{t_0}^{+*} $ is tangent to $C_{t_0}^- $ at some interior point. 

2.   $C_{t_0}^{+*} \geq C_{t_0}^-  $ and $C_{t_0}^{+*} $ and $C_{t_0}^- $ are tangent along their common boundary. 

Recall that $\Sigma$ is orthogonal to $\Gamma\times {\mathbb R}$ along $\mathcal{C}$, since $\mathcal{C}=\Sigma \cap (\Gamma\times {\mathbb R})$ and $\Sigma$ is symmetric with respect to $\Gamma\times {\mathbb R}$.

Thus, in the first case, applying the boundary maximum principle to the surfaces $\Sigma_{1,t_0}^{+*}$ and $\Sigma_{1,t_0}^-  $, we conclude that $\Sigma_{1,t_0}^{+*} = \Sigma_{1,t_0}^-  $ and then $\Sigma_{t_0}^{+*} = \Sigma_{t_0}^-  $. 

In the second case we apply the boundary maximum principle to the surfaces $\Sigma_{t_0}^{+*}$ and $\Sigma_{t_0}^- $ in order to infer $\Sigma_{t_0}^{+*} = \Sigma_{t_0}^-  $. 

Consequently, the surface $\Sigma$ is symmetric with respect to the horizontal plane $\Pi_{t_0}={\mathbb{H}^2} \times \{t_0\}$, as desired. 

It follows from the proof of Lemma \[L.symetrie-3\] that the tangent plane at any point of $\Sigma\setminus ({\mathbb{H}^2} \times \{t_0\}) $ is never vertical. 

The maximum principle shows that $\gamma_1 \not= \gamma_2$, since $\Sigma$ is not a vertical plane.  We know from Proposition \[P.non existence\] that ${\partial_{\infty}}\gamma_1 \cap {\partial_{\infty}}\gamma_2 =\emptyset$ and from Proposition \[P.non existence 2\] that $ \gamma_1 \cap \gamma_2 =\emptyset$. Thus, the geodesics $\gamma_1$ and $\gamma_2$ satisfy the properties (\[geodesiques disjointes\]).  Therefore, we deduce from Lemmas \[L.symetrie-1\], \[L.symetrie-2\] and \[L.symetrie-3\] that $\Sigma$ is symmetric with respect to the vertical planes $\gamma_0 \times {\mathbb R}$ and $\Gamma \times {\mathbb R}$ and also with respect to the slice $\Pi_0:={\mathbb{H}^2} \times \{0\}$ (up to a vertical translation). 

We call $S_0$ the reflection across the slice $\Pi_0$, $S_\Gamma$ the reflection across the vertical plane $\Gamma \times {\mathbb R}$ and $S_{\gamma_0}$ the reflection across the vertical plane $\gamma_0 \times {\mathbb R}$.

 

For any real number $s\not=0$, we denote by $\Gamma_s$ the equidistant line to $\Gamma$ with distance equal to $|s|$, which intersects $\gamma_0$ between $p_0$ and $p_1$ (resp.  $p_0$ and $p_2$) if $s>0$ (resp.  $s<0$).  We set $\Gamma_0=\Gamma$. For any $s\in {\mathbb R}$, we set $P_s:=\Gamma_s \times {\mathbb R}$.

We define $\Sigma^+:= \Sigma \cap ({\mathbb{H}^2} \times \mathclose] 0, +\infty\mathopen[).$ Since $\Sigma^+$ is a vertical graph, the tangent plane is never vertical along $\Sigma^+$. Consequently $\Sigma^+$ intersects any $P_s$ tranversally.  Since each component of $\Sigma^+ \setminus (\gamma_0\times {\mathbb R})$ is a horizontal graph with respect to $\Gamma$ and since $\Sigma$ is symmetric with respect to $\Pi_0$ and $\gamma_0\times {\mathbb R}$, we deduce that for any $s\in {\mathbb R}$ the intersection $\Sigma \cap P_s$ consists in a Jordan curve.  Therefore, $\Sigma$ is homeomorphic to an annulus.  Since $\Sigma$ has finite total curvature, we get that $\Sigma$ is conformally parametrized by ${\mathbb C}^*:={\mathbb C}\setminus \{0\}$.

Let $X: {\mathbb C}^* \rightarrow \Sigma \subset {\mathbb{H}^2} \times {\mathbb R}$ be a conformal parametrization of $\Sigma$. Thus, the symmetry $S_\Gamma$ corresponds to a anticonformal diffeomorphism $s_\Gamma : {\mathbb C}\cup\{\infty\} \rightarrow {\mathbb C}\cup\{\infty\}$ satisfying $s_\Gamma(0)=\infty$ and $s_\Gamma(\infty)=0$. Since the set of fixed points of $S_\Gamma$ in $\Sigma$ is a Jordan curve, the set of fixed points of $s_\Gamma$ is a circle $c_\Gamma$. Up to a conformal change of coordinates, we can assume that $c_\Gamma\subset{\mathbb C}$ is the unit circle centered at the origin.  Thus, we get $s_\Gamma (z)=1/{\overline{z}}$ for any $z\in {\mathbb C}^*$.

Then, we denote by $s_{\gamma_0} : {\mathbb C}\cup\{\infty\} \rightarrow {\mathbb C}\cup\{\infty\}$ the anticonformal diffeomorphism corresponding to $S_{\gamma_0}$. The set of fixed points of $s_{\gamma_0}$ in ${\mathbb C}^*$ is a straight line $L_\gamma$ passing to and punctured at the origin.  Up to a rotation we can assume that $L_\gamma=\{ {{\rm Re}}\, z=0\}$. Thus, we have $s_{\gamma_0}(z)=
-{\overline{z}}$ for any $z\in {\mathbb C}$.

At last, let us call $s_0 : {\mathbb C}\cup\{\infty\} \rightarrow {\mathbb C}\cup\{\infty\}$ the anticonformal diffeomorphism corresponding to $S_0$. The set of fixed points of $s_0$ in ${\mathbb C}^*$ is a straight line $L$ passing and punctured at the origin.  Since we have $(S_0 \circ S_{\gamma_0}) \circ (S_0 \circ S_{\gamma_0})= Id$ on $\Sigma,$ we must have $(s_0 \circ s_{\gamma_0}) \circ (s_0 \circ s_{\gamma_0})= Id$ on ${\mathbb C}^*$. Thus, $L$ must be orthogonal to $L_\gamma$ and we get $L=\{ {{\rm Im}}\, z=0\}$. Then, we have $s_0(z)= {\overline{z}}$ for any $z\in {\mathbb C}$.

We call $P_0^+$ the component of $({\mathbb{H}^2} \times {\mathbb R}) \setminus (\Gamma\times{\mathbb R})$ containing $\gamma_1 \times {\mathbb R}$ and we set $\Sigma^{++}:=\Sigma^+\cap P_0^+$. Finally, we call $\Sigma_0$ any of the two components of $\Sigma^{++}\setminus (\gamma_0
\times {\mathbb R})$. Thus, we recover the whole surface $\Sigma$ by applying the symmetries $S_0$, $S_\Gamma$ and $S_{\gamma_0}$ to the closure of $\Sigma_0$.

We can assume that $\Sigma_0$ is parametrized by the subset $$U_0:=\{z\in {\mathbb C}\mid |z| >1, \ {{\rm Re}}\, z<0,\ {{\rm Im}}\, z>0\}.$$

 

Since $\Sigma_0$ is simply connected, we can consider its conjugate $\Sigma_0^*$ which is a minimal surface in ${\mathbb{H}^2} \times {\mathbb R}$ uniquely defined up to an ambient isometry. 

From now on, for any object $x$ relative to $\Sigma_0$ we denote by $x^*$ the corresponding object relative to the conjugate surface $\Sigma_0^*$. Thus, $X^*:= (F^*,h^*):U_0 \rightarrow \Sigma_0^*\subset {\mathbb{H}^2}\times {\mathbb R}$ is a conformal parametrization of $\Sigma_0^*$.

Observe that the boundary of $\Sigma_0$ is composed of three arcs:

1.   A semi-complete curve $b_1 \subset (\gamma_0  \times{\mathbb R})$ with boundary point $q$.

2.   A compact arc $b_2 \subset (\Gamma \times {\mathbb R})$ with boundary $q$ and ${\widetilde}q$.

3.   A semi-complete curve $b_3\subset \Pi_0$ with boundary point ${\widetilde}q$.

In order to visualize the following discussion we consider the model of the unit disk for ${\mathbb{H}^2}$ (see Figure \[Fig. Conjugue\]). 

Up to an isometry we can assume that $q^*=0\in {\mathbb{H}^2}.$

Since $b_1$ is contained in a vertical plane, its conjugate $b_1^*$ must be a horizontal half-geodesic issue from $0$ (see [@Daniel end of Section 4.1]).  We can assume that ${\partial_{\infty}}b_1^* = \{i\}$.

Moreover, since $b_2$ is contained in a vertical plane, its conjugate $b_2^*$ must be a compact geodesic arc orthogonal to $b_1^*$ with endpoints ${\widetilde}q^*$ and $ 0.$ Finally, since $b_3$ is contained in a horizontal plane, its conjugate $b_3^*$ must be a vertical half-geodesic issue from $ {\widetilde}q^*$. We can assume that $b_3^*= {\widetilde}q^* \times \{t\geq 0 \}$.

We denote by $C\subset {\mathbb{H}^2}$ the geodesic passing through $ {\widetilde}q^*$, having $i$ in its asymptotic boundary and we denote by $C_0$ the segment of $C$ with boundary points ${\widetilde}q^*$ and $i.$ We call $\mathcal{D}$ the domain of ${\mathbb{H}^2}$ bounded by $b_1^*$, $b_2^*$ and $C_0$ such that ${\partial_{\infty}}\mathcal{D}=\{ i\}$.

![[]{ data-label="Fig. Conjugue"}](figure7)

In order to prove that $\Sigma$ is a two ends model surface, it is enough to prove that $\Sigma_0^*$ is a vertical graph over $\mathcal{D}$, with infinite data on $C_0$ and zero data on the two sides $b_1^*$ and $b_2^*$ (see [@P]). 

We call $\sigma $ the reflection in ${\mathbb{H}^2} \times {\mathbb R}$ across the vertical geodesic containing $b_3^*$.

Since $\Sigma_0^*$ is parametrized by $U_0$, the interior of ${\overline{\Sigma}}_0^* \cup \sigma(\Sigma_0^*) $ is parametrized by $${\widetilde}U_0 :=\big \{z\in {\mathbb C}\mid |z|>1,\ {{\rm Re}}\, z<0 \big\}.$$

We recall that $\phi^*=-\phi$, $h^*=-2\,{{\rm Re}}\, W$, since $h=2\,{{\rm Im}}\, W$ and $h^*$ is the harmonic conjugate of $h.$ Thus $W^*=-iW$. We may suppose that $h^* >0$ on ${\widetilde}U_0 $.  We are able to study the behavior of $F^*$ and $h^*$ on ${\widetilde}U_0 $ in the same way as we did for $F$ and $h$ in Section \[Sec. End\].  By Lemma \[L.Level set\] and Theorem \[T.Description\], we infer that $m=0$ for the end of $\Sigma$ parametrized by $\{ |z| >1\}$. We deduce that for each $\mu>0$ the level set $\{h^*=\mu\}$ is composed of complete curves, each one having a part with asymptotic direction the ray $\{ ri \mid r>0\}$ and the other part with asymptotic direction the ray $\{ -ri \mid r>0\}$. Each one of those level curves intersects $L^-:=\{{{\rm Im}}\, z=0 \} \cap  {\widetilde}U_0 $. 

 

Since ${\overline{\Sigma}}_0 \cup S_0(\Sigma_0)$ is a horizontal graph with respect to $\Gamma$, we get that the tangent plane to it is never horizontal.  Therefore, $\phi
\not=0$ on ${\widetilde}U_0$. Since $h\equiv 0$ on $L^-$, we deduce that $h^*$ is strictly monotonous on $L^-$. Therefore, for any $\mu>0$, the level set $\{h^*=\mu\}$ is composed of a unique complete curve $L_\mu$. Since $h$ is strictly monotonous along any curve $L_\mu$, we deduce that $W^*$ is one-to-one on ${\widetilde}U_0 $.  The results established in Theorem \[T.Description\] yields that the projections of the curves $L_\mu$ on ${\mathbb{H}^2}$ converge to the geodesic $C$ when $\mu \to +\infty$.

Let us call $B_i$ the geodesic containing $b_i^*$, $i=1,2$.

 

[*[**Claim 1.**]{}  We have $\Sigma_0^* \subset \mathcal{D} \times {\mathbb R}$.*]{}

By construction, $B_1$ and $B_2$ meet orthogonally at the origin and $B_2$ is the geodesic with asymptotic points $(-1,0),$ $(1,0).$ For $s>0$, we call $Q_s \subset {\mathbb{H}^2} \times {\mathbb R}$ the vertical plane orthogonal to $B_2$ at $(s,0)$ and we call $Q_s^-$ the component of $({\mathbb{H}^2} \times {\mathbb R})\setminus Q_s$ which does not contain $\mathcal{D} \times
{\mathbb R}$.

Observe that ${\partial_{\infty}}\Sigma_0^* = \{i\}\times {\mathbb R}$. Moreover, if $(z_n)$ is a sequence in $U_0$ such that $h^*(z_n)\to +\infty$, then we have $d_{{\mathbb{H}^2}}\big(F^* (z_n),C
\big)\to
0$. Consequently, for any $s>0,$ the intersection $\Sigma_0^* \cap Q_s^-$ is either empty or have compact closure.  Assume the latter is true.  In this case, we find $s$ such that $\Sigma_0^* \cap Q_s^- =\emptyset.$ Then, we start to decrease $s$. By the maximum principle, we can decrease $s$ till $0$ and obtain that $\Sigma_0^* \cap Q_s^- =\emptyset$ for any $s>0$. Therefore, $\Sigma_0^*$ remains in the component of $({\mathbb{H}^2} \times {\mathbb R})\setminus (B_1\times{\mathbb R})$ containing $\mathcal{D} \times {\mathbb R}$.

By the same reasoning as above we can prove that $\Sigma_0^*$ remains in the component of $({\mathbb{H}^2} \times {\mathbb R})\setminus (B_2\times{\mathbb R})$ containing $\mathcal{D} \times {\mathbb R}$ and also $\Sigma_0^*$ remains in the component of $({\mathbb{H}^2} \times {\mathbb R})\setminus (C\times{\mathbb R})$ containing $\mathcal{D} \times {\mathbb R}$. We conclude that $\Sigma_0^* \subset \mathcal{D} \times {\mathbb R}$.

 

[*[**Claim 2.**]{}  We have $F^* (U_0)\subset \mathcal{D}$. Furthermore, the map $F^* : U_0 \rightarrow \mathcal{D}$ is proper.*]{} 

Let $Pr : {\mathbb{H}^2} \times {\mathbb R}\rightarrow {\mathbb{H}^2}$ be the projection on the first component.  Since $F^*= Pr \circ X^*$ we deduce from Claim 1 that $F^* (U_0)\subset \mathcal{D}$.

We must prove that for any compact set $K\subset \mathcal{D}$, $(F^*)^{-1} (K)$ is a compact subset of $U_0$. In order to prove it, it is enough that for any sequence $(z_n)$ in $(F^*)^{-1} (K)$, there is a subsequence of $(z_n)$ converging in $(F^*)^{-1} (K)$.

Since $K$ is far from the geodesic $C$, the height function is bounded on $K$. Therefore, there exists a constant $\mu>0$ such that $(F^*)^{-1} (K)$ remains in the subset of $U_0$ bounded by the level set $L_\mu$ and the half-axis $\{iy \mid y>0 \}$. Let $(z_n)$ be a sequence in $(F^*)^{-1} (K)$. Suppose that $(z_n)$ is not bounded.  Therefore, there exists a subsequence $(z_{\varphi (n)})$ of $(z_n)$ such that $|z_{\varphi (n)}| \to +\infty$. As in the proof of Theorem \[T.Description\] (Assertion 2), we can show that there exists a sequence $(iy_n)$ such that $y_n \to +\infty$ and $d_{{\mathbb{H}^2}}\big(F^*(z_{\varphi (n)}), F^*(iy_n)\big)\to 0$. But this is a contradiction since we have, by construction, $F^*(z_{\varphi (n)}) \in K$ and $F^*(iy_n)\to i\in{\partial_{\infty}}{\mathbb{H}^2}$.

Thus $(z_n)$ is a bounded sequence of $U_0$. Therefore, since $U_0\subset {\mathbb C}$, we can extract a subsequence $(z_{\psi (n)})$ converging to some point $z\in {\overline{U}}_0$. We want to show that $z \in (F^*)^{-1} (K) $. 

Observe that $F^*$ maps the boundary of $U_0$ onto the boundary of $\mathcal{D}$, and moreover $d_{{\mathbb{H}^2}}\big( K,\partial\mathcal{D}\big)>0 $, since $K$ is a compact subset of $\mathcal{D}$. Therefore, we deduce that $z\in U_0$. Since $F^*$ is continuous and $K$ is compact, we obtain $F^*(z) \in K$, from which we get $z \in (F^*)^{-1} (K) $.  Therefore $F^* : U_0 \rightarrow \mathcal{D}$ is a proper map, as desired. 

[*[**Claim 3.**]{}  We have $F^* (U_0)= \mathcal{D}$ and $\Sigma_0^*$ is a vertical graph over $\mathcal{D}$.*]{}

We know from Claim 2 that $F^* (U_0) \subset\mathcal{D}$. Therefore, it suffices to prove that $F^* (U_0)$ is a closed and open subset of $\mathcal D.$

It is known that $n_3^*= \pm n_3$ (see [@HST Remark 9]).  On the other hand, since $\Sigma_0$ is a vertical graph, we have $n_3 \not=0$ along $\Sigma_0$. We deduce that the tangent plane is never vertical along $\Sigma_0^*$ and that the map $F^* : U_0 \rightarrow  \mathcal{D}\subset {\mathbb{H}^2} $ is a local smooth diffeomorphism.  Then, $F^* $ is an open map.  As $U_0$ is open, we get that $F^* (U_0)$ is an open subset of $\mathcal{D}$.

Now we prove that $F^* (U_0)$ is also a closed subset of $\mathcal D.$

Let $(q_n)$ be a sequence in $F^* (U_0)$ converging to some point $q\in\mathcal{D}$. We want to prove that $q\in F^* (U_0)$.

We set $K:=\{q\} \cup \{q_n,\ n \in \n\}$, then $K$ is a compact subset of $\mathcal{D}$. From Claim 2 we deduce that $(F^*)^{-1}(K)$ is a compact subset of $U_0$. For any $n\in \n$ there exists $z_n \in (F^*)^{-1}(K)$ such that $F^*(z_n)=q_n$. Therefore, we can extract a subsequence $(z_{\varphi (n)})$ which converges to some $z\in (F^*)^{-1}(K)\subset U_0$. Since $F^*$ is continuous we obtain $F^*(z_{\varphi (n)}) \to F^* (z)$, that is $q_{\varphi (n)} \to F^*(z)$. We deduce that $q=F^*(z)$ and then $q\in F^*(U_0)$, therefore $F^* (U_0)$ is a closed subset of $\mathcal D.$ Consequently we get $F^* (U_0)=\mathcal{D}$.

Hence, the map $F^* : U_0 \rightarrow  \mathcal{D}$ is a local smooth diffeomorphism.  Moreover $F^*$ is proper and surjective.  We deduce that it is a covering map.  Since $\mathcal{D}$ is connected and simply connected and $U_0$ is connected, we deduce that $F^*$ is a global diffeomorphism from $U_0$ onto $\mathcal{D}$, that is $\Sigma_0^*$ is a vertical graph over $\mathcal{D}$, as desired. 

Appendix.  Basic geometry in ${\mathbb{H}^2}$
============================================

In this section, we establish some background material about $C^2$-curves in ${\mathbb{H}^2}$ whose absolute value of geodesic curvature is strictly smaller than one.  We observe that the condition on the curvature implies that such curve is embedded, see for example [@ST3 Proposition 2.6.32]. 

\[P.limit\] Let $c:[0,+\infty\mathclose[ \rightarrow {\mathbb{H}^2}$ be a regular $C^2$-curve with infinite length.  Let $\kappa (t)$ be the geodesic curvature of $c$ at the point $c(t)$ and assume $\rvert \kappa(c(t)) \rvert < k <1$, for any $t\geq 0$. Then, the curve $C:= c([0,+\infty\mathclose[)$ has no limit point in ${\mathbb{H}^2},$ and the asymptotic boundary of $C$ consists in only one point $\{p_\infty \}={\partial_{\infty}}C $. 

$   $ If $k=0$ then $C$ is a part of a geodesic and the assertions are obvious.  Therefore we assume that $0<k<1$.

Indeed, assume by contradiction that there exists $p\in {\mathbb{H}^2}$ and a sequence of positive numbers $(t_n)$ such that $t_n \xrightarrow[n\to \infty]{} +\infty$ and $p_n := c(t_n) \xrightarrow[n\to \infty]{} p$.

Assume first that there exists a point $q\in C$, $q=c(t_0)$ for some $t_0>0$, such that $C$ is orthogonal at $q$ to the geodesic passing through $q$ and $p$. Let $H_q\subset {\mathbb{H}^2}$ be the horocycle through $q,$ tangent to the curve $C,$ such that $p$ belongs to the convex component of ${\mathbb{H}^2} \setminus H_q$. Noticing that $\rvert \kappa (c(t))\rvert <1 $, and the absolute value of the curvature of the horocycles is 1.  Thus, the maximum principle for curves, see [@ST3 Theorem 2.6.27], ensures that $C_0 := c([t_0, +\infty\mathclose[)$ belongs to the nonconvex component of ${\mathbb{H}^2} \setminus H_q$ and then, $p$ cannot not be in the closure of $C$. 

Hence we infer that the function $t \mapsto d_{{\mathbb{H}^2}}(c(t), p)$ is strictly decreasing. 

For $t>0$ we denote by $\alpha(t)\in[0,\pi]$, the nonoriented angle at $c(t)$ between the tangent vector $c^\prime (t)$ and the geodesic segment $[c(t), p]$. Since the function $t \mapsto d_{{\mathbb{H}^2}}(c(t), p)$ is strictly decreasing we have that $\alpha(t)\in[0,\pi/2[$ for any $t>0$.

Actually, we have $\alpha (t) \rightarrow 0$ as $t \to +\infty$. Indeed, assume by contradiction that there exist a sequence $(t_n)$, and a real number $ \alpha_0\in(0,\pi/2)$, such that $t_n\to +\infty$ and $\alpha (t_n)\to \alpha_0$. For any $n\in \n$, we denote by $\gamma_n$ the geodesic of ${\mathbb{H}^2}$ through $c(t_n)$ tangent to $C$.  Let us denote by $H_{n}\subset{\mathbb{H}^2}$, $n\in \n$, the horocycle through $c(t_n)$ tangent to the curve $C$ and contained in the same component of ${\mathbb{H}^2} \setminus \gamma_n$ as $p$. Therefore, for $n$ large enough, the point $c(t_n)$ is very close to $p$ and the angle $\alpha(t)$ is very close to $\alpha_0$. This would imply, for $n$ large enough, that $p$ belongs to the convex component of ${\mathbb{H}^2} \setminus H_n$ and this would give again a contradiction with the maximum principle for curves.  Therefore we get that $\alpha(t)\to 0$ as $t\to +\infty$.

To conclude the argument, we choose for ${\mathbb{H}^2}$ the model of the unit disk equipped with the metric $g_{{\mathbb D}}=\lambda^2 (z)\,\rvert dz \rvert^2$, where $\lambda (z)=2/(1-\rvert z\rvert^2)$. We can assume that $p=0$ and that $C$ is parametrized by arclength. 

In polar coordinates we have $c(t)=(r(t)\cos \theta(t), r(t)\sin \theta(t))$ where $r(t)=\rvert c(t)\rvert >0$ and $\theta (t)\in {\mathbb R}$. We set $\partial_r := (\cos \theta,\sin \theta)$. Since $\alpha(t)\to 0$, we have $$\frac{\langle c^\prime (t)\, ;\, \partial_r\rangle_{\mathbb D}}
{\rvert\partial_r\rvert_{\mathbb D}} \to -1,$$ where the scalar product and the norm are considered with respect to the metric $g_{\mathbb D}$. From which we get that $\lambda (c(t)) r^\prime (t) \to -1$. Using that $c(t)\to p=0$ for $t\to\infty,$ we obtain $ \lambda (c(t))\to 2$ for $t\to\infty$, therefore $r^\prime (t) \to -1/2$ and then $r(t)\to -\infty$ as $t\to +\infty$. This is a contradiction and this concludes the proof of the claim. 

 

Since $C$ is embedded and has no limit point in ${\mathbb{H}^2}$, we deduce that its asymptotic boundary is not empty.  Let $p_\infty \in {\partial_{\infty}}C$ be an asymptotic point of $C$. 

[*[**Claim 2.**]{}  $p_\infty$ is the unique asymptotic point of $C$.*]{} 

Let $\Gamma \subset {\mathbb{H}^2}$ be a complete curve with constant curvature $k$ such that $p_\infty \not\in {\partial_{\infty}}\Gamma$ and $p_\infty$ belongs to the asymptotic boundary of the convex component of ${\mathbb{H}^2} \setminus  \Gamma$.

Let $\gamma \subset {\mathbb{H}^2}$ be a geodesic intersecting $\Gamma$ such that $p_\infty \in {\partial_{\infty}}\gamma$. For any $s\in {\mathbb R}$, let $\Gamma_s$ be the translated copy of $\Gamma$ along $\gamma$ at distance $\rvert s \rvert$, towards $p_\infty$ if $s>0$, and in the opposite direction otherwise.  We denote by $\Gamma_s^+$ the convex component of ${\mathbb{H}^2} \setminus \Gamma_s$. Thus, we have $\Gamma_0=\Gamma$ and $p_\infty \in {\partial_{\infty}}\Gamma_s^+$ for any $s \geq 0$. Observe that $\cap_{s \geq 0} {\partial_{\infty}}\Gamma_s^+ =\{ p_\infty\}$.

If we assume that for any $s >0$, there exists $t_s >0$ such that $c([t_s, +\infty\mathclose[) \subset {\overline{\Gamma_s^+}},$ then we deduce that $p_\infty$ is the unique asymptotic point of $C,$ as desired.  Therefore, we are left with the proof of such assumption. 

Suppose by contradiction that there exists $r >0$ such that for any $t >0$ the curve $c([t, +\infty\mathclose[)$ is not entirely contained in ${\overline{\Gamma_{r}^+}}$. Therefore, there is an arc $C_1\subset C$ such that $\partial C_1 \subset \Gamma_r$ and $C_1 \cap  \Gamma_{r}^+ =\emptyset$, that is $C_1$ stays outside $\Gamma_{r}^+$. Note that ${\overline{C}}_1$ is a compact arc with boundary on $\Gamma_r$. Considering the curves $\Gamma_s$, for $s$ going from $r$ to $-\infty$, we get a real number $\rho < r$ such that $C_1 \subset {\overline{\Gamma_{\rho}^+}}$ and $C_1$ and $ \Gamma_{\rho}$ are tangent at some interior point of $C_1$. This gives a contradiction by the maximum principle, keeping in mind the hypothesis about the curvature of $C$ and $\Gamma_\rho$ and the fact that $C_1$ belongs to the closure of the convex component of ${\mathbb{H}^2} \setminus \Gamma_\rho$. This concludes the proof . 

\[D.C1\] [(1)]{} Let $(p_n)$ be a sequence in ${\mathbb{H}^2}$ converging to some point $p\in {\mathbb{H}^2}$. Let $v\in T_p{\mathbb{H}^2}$ and $v_n \in T_{p_n} {\mathbb{H}^2}$ be non zero vectors.  Assuming that $p_n\not=p$, we denote by $c_n$ the geodesic passing through $p$ and $p_n$, and by $T_n$ the translation along $c_n$ such that $T_n (p)=p_n$. If $p_n=p$, we set $T_n=Id$. Let $\alpha_n\in [0,\pi]$ be the non-oriented angle between $v_n$ and $T_n (v)$.

We say that the sequence $(v_n)$ [*converges*]{} to $v$ (denoted shortly by $v_n \rightarrow v$) if $\alpha_n \rightarrow 0$ and $|v_n|_{{\mathbb{H}^2}} \to |v|_{{\mathbb{H}^2}}$.

[(2)]{} Let $\gamma$ be a geodesic in ${\mathbb{H}^2}$ and $(\gamma_n)$ be a sequence of complete and regular $C^1$-curves in ${\mathbb{H}^2}.$ We say that the sequence $(\gamma_n)$ [*converges to $\gamma$ in the $C^1$ topology* ]{} if:

1.   \[C0\] For any $\varepsilon >0$, there is $n_0\in \n$ such that for any $n>n_0$ the curve $\gamma_n$ stays in the region of ${\mathbb{H}^2}$ bounded by the two equidistant lines of $\gamma$ with distance $\varepsilon$ from $\gamma$.

2.   \[C1\] Let $p\in \gamma$, and let $(p_n)$, $p_n\in \gamma_n$, be any sequence converging to $p$. Let $v_n\in T_{p_n}\gamma_n$ be a unit tangent vector of $\gamma_n$ at $p_n$. If the sequence $v_n$ converges to a unit vector $v\in T_p{\mathbb{H}^2}$, then $v\in T_p \gamma$, that is $v$ is tangent to $\gamma$ at $p$.

\[P.C1\] Let $\gamma  \subset {\mathbb{H}^2}$ be a geodesic .  Let $(\gamma_n)$ be a sequence of complete and regular $C^2$-curves such that:

-   ${\partial_{\infty}}\gamma_n = {\partial_{\infty}}\gamma$ for any $n \in \n$.

-   $\sup _{q\in \gamma_n}\big\{\rvert \kappa_{\gamma_n}(q)
    \rvert \big\}
    \xrightarrow[n\to \infty]{} 0 $, where $ \kappa_{\gamma_n}(q) $ is the geodesic curvature of $\gamma_n$ at point $q\in \gamma_n$.

Then, the sequence $(\gamma_n)$ converges to $\gamma$ in the $C^1$ topology. 

Let us prove [*(i)*]{} of Definition \[D.C1\]. 

Let $\varepsilon >0$ and let $n_0\in \n$ such that for any $n > n_0$ we have $$\tanh \varepsilon > \sup _{q\in \gamma_n}\big\{\rvert \kappa_{\gamma_n}(q)
\rvert
\big\}.$$ Denote by $L_\varepsilon ^1$ and $L_\varepsilon ^2$ the two equidistant lines of $\gamma$ with distance $\varepsilon$ from $\gamma$. It suffices to show that, for any $n>n_0$, the curve $\gamma_n$ belongs to the convex component of both ${\mathbb{H}^2} \setminus L_\varepsilon ^1$ and ${\mathbb{H}^2} \setminus L_\varepsilon ^2$. We prove that fact for $L_\varepsilon :=L_\varepsilon ^1$. The proof for $L_\varepsilon ^2$ is analogous. 

 

Let $\gamma^\prime$ be any geodesic of ${\mathbb{H}^2}$ different from $\gamma$, intersecting $\gamma$. Let $p_\infty ^\prime\in {\partial_{\infty}}{\mathbb{H}^2} $ be the asymptotic point of $ \gamma^\prime$ which stays in the asymptotic boundary of the nonconvex component of ${\mathbb{H}^2}\setminus L_\varepsilon$. Consider the translations along $ \gamma^\prime,$ towards $p_\infty ^\prime$. Assume by contradiction that for any $n_0$ there exists $n>n_0$ such that $\gamma_n$ does not belong to the convex component of ${\mathbb{H}^2} \setminus L_\varepsilon$, there exists a translated copy $L_\varepsilon^\prime $ of $L_\varepsilon$ such that:

-   $L_\varepsilon^\prime$ intersects the curve $\gamma_n$ at one point $q_n$.

-   $L_\varepsilon^\prime$ and $\gamma_n$ are tangent at $q_n$.

-   $\gamma_n$ belongs to the closure of the convex component of ${\mathbb{H}^2}\setminus L_\varepsilon^\prime$.

Since the geodesic curvature of $L_\varepsilon^\prime$ is $\tanh \varepsilon$ (with respect to the normal direction pointing towards the convex component of ${\mathbb{H}^2}\setminus L_\varepsilon^\prime$) and since $\tanh \varepsilon >\sup _{q\in \gamma_n}\big\{\rvert \kappa_{\gamma_n}(q)
\rvert
\big\}$, we obtain a contradiction with the maximum principle.  This completes the proof of [*(i)*]{}. 

Now, we prove [*(ii)*]{} of Definition \[D.C1\]. 

By contradiction, assume that the unit vector $v\in T_p {\mathbb{H}^2}$ is not tangent to $\gamma$.

Let $\varepsilon >0$ and let $L\subset {\mathbb{H}^2}$ be one of the two complete curves passing through $p$, tangent to $v$ whose absolute value of the geodesic curvature is $\tanh \varepsilon$. Since $v$ is not tangent to $\gamma$, if $\varepsilon$ is small enough, then the curve $L$ separates the two points of the asymptotic boundary of $\gamma$, say $p_\infty$ and $q_\infty$.

As in the Definition \[D.C1\], we denote by $c_n$ the geodesic passing through $p$ and $p_n$. Let $T_n$ be the hyperbolic translation along $c_n$ such that $T_n(p)=p_n$. Let $R_n$ be the rotation in ${\mathbb{H}^2}$ around $p_n$ such that $R_n ( T_n(v))=v_n$. Therefore, $L_n :=(R_n \circ T_n)(L)$ is a complete curve through $p_n$, tangent to $\gamma_n$ at $p_n$, with constant (absolute value) curvature equal to $\tanh \varepsilon$. If $n$ is large enough then the curve $L_n$ separates $p_\infty$ and $q_\infty$, since this is true for $L$.

Observe that, if $n$ is large enough, we have $\sup _{q\in \gamma_n}\big\{\rvert \kappa_{\gamma_n}(q) \rvert \big\}
<\tanh \varepsilon$. Consequently, using the maximum principle for curves in the same way as before, we can show that $\gamma_n$ entirely belongs to the closure of one of the two components of ${\mathbb{H}^2} \setminus L_n$. But this gives a contradiction with the assumption that ${\partial_{\infty}}\gamma_n = {\partial_{\infty}}\gamma=\{p_\infty, q_\infty\}$. We conclude that $v$ is tangent to $\gamma$, as desired. 

We can extend Definition \[D.C1\] to any dimensional hyperbolic space ${\mathbb{H}^n},\ n\geq 2$. Moreover, we can prove in the same way as in Proposition \[P.C1\], that if $\Pi\subset {\mathbb{H}^n}$ is a geodesic hyperplane and if $(\Pi_n)$ is a sequence of complete and regular $C^2$-hypersurfaces of ${\mathbb{H}^n}$ such that ${\partial_{\infty}}\Pi_n={\partial_{\infty}}\Pi$ for any $n$ and $\sup_{q \in \Pi_n}\{| H_n (q)|\}\to 0 $, where $H_n(q)$ denotes the mean curvature of $\Pi_n$ at $q$, then the sequence $(\Pi_n)$ converges $C^1$ to $\Pi$.

\[P.graph\] Let $\gamma_1 \subset {\mathbb{H}^2}$ be a geodesic and let $p_\infty\in {\partial_{\infty}}{\mathbb{H}^2}$ such that $p_\infty\not\in  {\partial_{\infty}}\gamma_1$.

For any $\rho >0$, let $L_\rho$ be the equidistant line to $\gamma_1$ whose distance to $\gamma_1$ is $\rho$, such that $p_\infty$ belongs to the asymptotic boundary of the nonconvex component of ${\mathbb{H}^2}\setminus L_\rho$.

Let $0< k <1$ and let $c : [0,\infty\mathclose[ \rightarrow {\mathbb{H}^2}$ be a regular $C^2$-curve such that ${\partial_{\infty}}c([0,\infty\mathclose[ )= \{p_\infty\}$ and such that $\rvert \kappa (c(t))\rvert  <k$ for any $t\geq 0$. Set $\rho_0=\max \{d_{{\mathbb{H}^2}}\big(c(0),\gamma_1\big), \tanh^{-1} (k)\}$.

Then, for any $\rho > \rho_0$, one has the following facts. 

1.   The equidistant line $L_\rho$ cuts the curve $c([0,\infty\mathclose[ )$ at a unique point.  Therefore, there exists $t_0>0$ such that the curve $c([t_0,\infty[)$ is a horizontal graph with respect to $\gamma_1.$

2.   The equidistant line $L_\rho$ is transversal to $c([0,\infty\mathclose[ ).$

Let $\rho >\rho_0$ and let $C= c\big([0,\infty\mathclose[ \big)$. Since ${\partial_{\infty}}C=p_\infty$, the equidistant line $L_\rho$ must intersect the curve $C$ at least at one point.  Assume by contradiction that $L_\rho$ cut $C$ at two or more points.  By construction, $c(0)$ belongs to the convex component of ${\mathbb{H}^2} \setminus L_\rho$. Let $p_1\in  L_\rho \cap C$ be the first intersection point from $c(0)$. The boundary maximum principle for curves shows that the curves $ L_\rho $ and $C$ are not tangent at $p_1$. Let $p_2\in  L_\rho \cap C$ be the first intersection point after $p_1$. Thus, the whole arc of $C$ between $p_1$ and $p_2$ belongs to the nonconvex component of ${\mathbb{H}^2} \setminus L_\rho$. Now we obtain a contradiction with the maximum principle in the following way. 

Let $p_\infty^\prime\in {\partial_{\infty}}{\mathbb{H}^2}$ be a point in the asymptotic boundary of the convex component of ${\mathbb{H}^2} \setminus  L_\rho$. Let $\gamma\subset {\mathbb{H}^2}$ be the geodesic such that ${\partial_{\infty}}\gamma =\{p_\infty, p_\infty^\prime\}$. Considering the hyperbolic translations along $\gamma$ towards $p_\infty$, we obtain a translated copy $L_\rho^\prime$ of $L_\rho$ such that:

-   $L_\rho^\prime$ intersects the arc of $C$ between $p_1$ and $p_2$ at one point $q$.

-   $L_\rho^\prime$ and the arc of $C$ between $p_1$ and $p_2$ are tangent at $q$.

-   The arc of $C$ between $p_1$ and $p_2$ belongs to the closure of the convex component of ${\mathbb{H}^2}\setminus L_\rho^\prime$.

Since the geodesic curvature of $L_\rho^\prime$ is $\tanh \rho$ (with respect to the normal direction pointing towards the convex component of ${\mathbb{H}^2}\setminus L_\rho^\prime$) and since $\tanh \rho > k > \sup_{q\in C}\{\rvert \kappa (C) \rvert\} $, we obtain a contradiction with the maximum principle. 

So Assertion (1) is proved.  Now we prove Assertion (2). 

Suppose, by contradiction, that for some $\rho > \rho_0$, the equidistant line $L_{\rho}$ is tangent to the curve $C$ at some point $p_1$. We set $C_1=c\big(\mathopen] \rho_0, \infty\mathclose[\big)$. Then, the first part of the proof shows that the curve $C_1\setminus \{p_1\}$ remains in the nonconvex component of ${\mathbb{H}^2} \setminus L_{\rho}$. Therefore, for $\varepsilon >0$ small enough, the equidistant line $L_{\rho + \varepsilon }$ intersects the curve $C$ at least at two different points near $p_1$, giving a contradiction with assertion (1). 

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[^1]: Mathematics subject classification: 53A10, 53C42, 49Q05.\
    The second, third and fourth authors were partially supported by CNPq and FAPERJ of Brasil.  The second author was partially supported by PRIN 2007: Sottovarietá, strutture speciali e geometria asintotica.