Height estimate for special Weingarten surfaces of elliptic type in ${\mathbb{M}}^2(c) \times \mathbb{R}$

Abstract

In this article we provide a vertical height estimate for compact special Weingarten surfaces of elliptic type in ${\mathbb{M}}^2(c) \times \mathbb{R}$, i.e. surfaces whose mean curvature $H$ and extrinsic Gauss curvature $K_e$ satisfy $H=f(H^2-K_e)$ with $4x(f'(x))^2<1,$ for all $x \in [0,+\infty ).$ The vertical height estimate generalizes a result by Rosenberg and Sa Earp and applies only to surfaces verifying a height estimate condition. When $c<0,$ using also a horizontal height estimate, we show a non-existence result for properly embedded Weingarten surfaces of elliptic type in $\mathbb{H}^2(c) \times \mathbb{R}$ with finite topology and one end.

1. Introduction

In this work we will consider special Weingarten surfaces of elliptic type in ${\mathbb{M}}^2(c) \times {\mathbb{R}}.$ Here ${\mathbb{M}}^2(c)={\mathbb{S}}^2(c),{\mathbb{R}}^2$ or ${\mathbb{H}}^2(c)$ depending on the sign of the sectional curvature $c.$ If $H$ and $K_e$ denote the mean curvature and the extrinsic Gauss curvature of a surface $\Sigma$ respectively, then $\Sigma$ is called a special Weingarten surface if the following identity holds:

$$\begin{equation} H=f(H^2-K_e), {\tag{1}} \end{equation}$$

with $f\in {\mathcal{C}}^0([0,+\infty )).$ Furthermore if $f\in {\mathcal{C}}^1([0,+\infty ))$ and $4x(f'(x))^2<1 \, \forall x\in [0,+\infty ),$ then $f$ is said to be elliptic and $\Sigma$ is said to be a special Weingarten surface of elliptic type, henceforth called a SWET surface.

The study of Weingarten surfaces started with H. Hopf 7, P. Hartman and W. Wintner 8 and S. S. Chern 3, who considered compact Weingarten surfaces in ${\mathbb{R}}^3$. H. Rosenberg and R. Sa Earp 14 showed that compact special Weingarten surfaces in ${\mathbb{R}}^3$ and ${\mathbb{H}}^3$ satisfy an a priori height estimate, assuming also that $f$ satisfies a height estimate condition, and used this fact to prove that the annular ends of a properly embedded special Weingarten surface $M$ are cylindrically bounded. Moreover, if such an $M$ is non-compact and has finite topological type, then $M$ must have more than one end; if $M$ has two ends, then it must be a rotational surface; and if $M$ has three ends, it is contained in a slab. They followed the ideas of Meeks 11 and Korevaar-Kusner-Solomon 10 for non-zero constant mean curvature surfaces in ${\mathbb{R}}^3$. Recently, Aledo-Espinar-Gálvez in 2 obtained a geometric height estimate for SWET surfaces with $f(0)\neq 0$ in ${\mathbb{M}}^3(c),$ $c \leq 0,$ with no other hypothesis on $f.$

R. Sa Earp and E. Toubiana in 161718 studied rotational special Weingarten surfaces in ${\mathbb{R}}^3$ and ${\mathbb{H}}^3$. In the case $f(0)\neq 0$ (constant mean curvature type), they determined necessary and sufficient conditions for existence and uniqueness of examples whose geometric behaviour is the same as the one of Delaunay surfaces in ${\mathbb{R}}^3$, i.e. unduloids (embedded) and nodoids (non-embedded), which have non-zero constant mean curvature. In the case $f(0)=0$ (minimal type), they established the existence of examples whose geometric behaviour is the same as those of the catenoid of ${\mathbb{R}}^3$, which is the only rotational minimal surface in ${\mathbb{R}}^3$.

By arguments similar to those used by Sa Earp and Toubiana, the author and M. Rodriguez in 12 determined necessary and sufficient conditions for existence and uniqueness of rotational SWET surfaces in $\mathbb{S}^2\times {\mathbb{R}}$ and ${\mathbb{H}}^2 \times {\mathbb{R}}$ of minimal type ($f(0)=0$).

The reason we focus on SWET surfaces is that the ellipticity of $f$ ensures that the operator obtained by linearization of 1 is elliptic in the sense of Hopf 7 and solutions to 1 satisfy an interior and a boundary maximum principle.

Furthermore we show that an estimate for the height (defined below), similar to one given in 14, holds for SWET surfaces in product manifolds of dimension three under additional assumptions of $f$.

Let $\Sigma$ be a connected orientable hypersurface immersed in ${\mathbb{M}}^2(c) \times {\mathbb{R}}.$ The height function, denoted by $h,$ of $\Sigma$ is defined as the restriction to $\Sigma$ of the projection $t: {\mathbb{M}}^2(c) \times {\mathbb{R}}\to {\mathbb{R}}.$

Theorem 1.1 (Height estimate).

Let $\Sigma$ be a compact SWET surface embedded in ${\mathbb{M}}^2(c) \times {\mathbb{R}}$ which is a graph over ${\mathbb{M}}^2(c) \times \{0\}$ with $\partial \Sigma \subset {\mathbb{M}}^2(c) \times \{0\}.$ Let $x=H^2-K_e.$ If $f>0,$ $f-2xf'>0$ and $f^2+c+x(1-4ff') >0,$ then

$$|h|\leqslant 1/l$$

where

$$\begin{equation} l=\min _\Sigma \, \left( 2f+ \frac{c-K_e}{f(1-2ff')+2K_e f'} \right). {\tag{2}} \end{equation}$$

Corollary 1.2 (Height estimate for cmc surfaces).

Let $\Sigma$ be a compact surface having constant mean curvature which is embedded in ${\mathbb{M}}^2(c) \times {\mathbb{R}}$ and a graph over ${\mathbb{M}}^2(c) \times \{0\}$ such that $\partial \Sigma \subset {\mathbb{M}}^2(c) \times \{0\}.$ Suppose that

(1)

$c \leq 0,$ $H> \sqrt {\frac{\max _\Sigma K_e -c}{2}},$ or

(2)

$c>0,$ $\max _\Sigma K_e -c > 0,$ $H> \sqrt {\frac{\max _\Sigma K_e -c}{2}},$ or

(3)

$c>0,$ $\max _\Sigma K_e -c \leq 0,$ $H> 0;$

then

$$|h| \leqslant \frac{H}{2H^2+c-\max _\Sigma K_e}.$$

The technique used in 2 to prove a geometric height estimate for SWET surfaces in $\mathbb{M}^3(c)$, $c \leq 0,$ does not apply to our setting.

Theorem 6.2 in 6 provides a horizontal height estimate for compact surfaces $\Sigma$ with constant curvature or constant mean curvature in $\mathbb{H}^2(c) \times {\mathbb{R}}$ and boundary contained in a vertical plane. The case of surfaces in ${\mathbb{S}}^2(c) \times {\mathbb{R}}$ is not considered to be ${\mathbb{S}}^2(c)$ compact.

It is possible to prove a similar estimate for SWET surfaces.

Theorem 1.3 (Horizontal height estimate).

Let $P$ denote a vertical plane in ${\mathbb{H}}^2(c) \times {\mathbb{R}}.$ Let $\Sigma$ be a compact SWET surface in ${\mathbb{H}}^2(c) \times {\mathbb{R}},$ with $\partial \Sigma \subset P.$ Assume that the elliptic function $f \geq H_0 > \frac{\sqrt {-c}}{2}.$ Then for every $p\in \Sigma$, the horizontal distance in ${\mathbb{H}}^2(c) \times {\mathbb{R}}$ of $p$ to $P$ is bounded by a constant $C$ which does not depend on $\Sigma$.

The proof of Theorem 6.2 in 6 applies verbatim to our setting, with a unique exception: the proof uses the maximum principle to compare $\Sigma$ to a surface $\Sigma _0$ that in our case has to be the sphere of constant mean curvature equal to $H_0.$

Combining Theorems 1.1 and 1.3 we are in order to prove the following non-existence result.

Theorem 1.4.

There are no properly embedded SWET surfaces in ${\mathbb{H}}^2(c) \times {\mathbb{R}}$ with finite topology, one end and whose elliptic function $f$ satisfies the hypotheses of Theorems 1.1, 1.3.

Such a theorem generalizes Theorem 7.2 of 6, which applies to surfaces with constant mean curvature $H>1/2$ or constant curvature $K>0$ which are properly embedded in ${\mathbb{H}}^2 \times {\mathbb{R}}.$

2. Proof of Theorem 1.1

2.1. Preliminaries

Let $\Sigma$ be a oriented connected Riemannian $m$-manifold and let $F:\Sigma \to {\mathbb{M}}^{m+1}$ be an isometric immersion of $\Sigma$ into an orientable Riemannian $(m+1)$-manifold ${\mathbb{M}}^{m+1}.$ We choose a normal unit vector field $N$ along $\Sigma$ and define the shape operator $A$ associated with the second fundamental form of $\Sigma$; that is, for any $p \in \Sigma ,$

$$\langle A(X),Y \rangle = -\langle \overline{\nabla }_X N,Y \rangle ,\, X, Y \in T_p \Sigma ,$$

where $\overline{\nabla }$ is the Riemannian connection of ${\mathbb{M}}^{m+1}.$

Let $k_1,\ldots ,k_m$ denote the eigenvalues of $A.$ For $1 \leqslant r \leqslant m,$ let $S_r$ denote the $r$-th symmetric function of $k_1,\ldots ,k_m$ and $T_r$ be the $r$-th Newton transformation: $T_0=I,$ $T_r=S_rI-AT_{r-1}.$

If $H_r$ denotes the $r$-th mean curvature of $\Sigma ,$ then $H_r=S_r/C^r_m,$ where $C^r_m=\frac{m!}{r!(m-r)!}.$

Let us consider a domain $D \subset \Sigma$ such that its closure $\overline{D}$ is compact with smooth boundary.

Definition 2.1.

A variation of $D$ is a differentiable map $\phi :(-{\varepsilon },{\varepsilon }) \times \Sigma \to {{\mathbb{M}}}^{m+1},$ where ${\varepsilon }>0,$ such that for each $s \in (-{\varepsilon },{\varepsilon })$ the map $\phi _s: \Sigma \to {{\mathbb{M}}}^{m+1}$ defined by $\phi _s(p)=\phi (s,p)$ is an immersion and $\phi _0(p)=F(p)$ for every $p \in \Sigma$ (we recall that $F$ denotes the immersion of $\Sigma$ in ${{\mathbb{M}}}^{m+1}$) and $\phi _s(p)=F(p)$ for $p \in \Sigma \setminus \overline{D}$ and $s \in (-{\varepsilon },{\varepsilon })$.

We set

$$E_s(p)=\frac{\partial \phi }{\partial s}(s,p) \,\quad \text{and} \, f_s=\langle E_s,N_s\rangle ,$$

where $N_s$ is the unit normal vector field along $\phi _s(\Sigma ).$ $E$ is called the variational vector field of $\phi .$ Let $A_s(p)$ be the shape operator of $\phi _s(\Sigma )$ at the point $p$ and $S_r(s,p)$ the $r$-th symmetric function of the eigenvalues of $A_s(p).$

Definition 2.2.

Let $g \in {\mathcal{C}}^2(\Sigma ).$ We define $L_r (g)=div(T_r \nabla g),$ $0\leqslant r \leqslant m.$

In 5 M.F. Elbert proved that, for $1\leqslant r \leqslant m,$

$$\begin{equation} \frac{\partial S_r}{\partial s}= L_{r-1}(f_s)+ f_s(S_1S_r-(r+1)S_{r+1})+f_s tr(T_{r-1} \overline{R}_N)+E^T_s(S_r), {\tag{3}} \end{equation}$$

where $\overline{R}_N$ is defined as $\overline{R}_N(X)=\overline{R}(N,X)N,$ $\overline{R}$ is the curvature tensor of ${\mathbb{M}}^{m+1}$ and $E_s^T$ denotes the tangent part of $E_s.$

In the sequel we will consider the case where ${\mathbb{M}}^{m+1}$ has a special structure: ${\mathbb{M}}^{m+1}={\mathbb{M}}^m \times {\mathbb{R}},$ where ${\mathbb{M}}^m$ is an $m$-dimensional Riemannian manifold.

Definition 2.3.

Let $\Sigma$ be a connected orientable hypersurface immersed in ${\mathbb{M}}^m \times {\mathbb{R}}.$ The height function, denoted by $h,$ of $\Sigma$ in ${\mathbb{M}}^m \times {\mathbb{R}}$ is defined as the restriction to $\Sigma$ of the projection $t: {\mathbb{M}}^m \times {\mathbb{R}}\to {\mathbb{R}}.$

The following result has been proved in 4 by X. Cheng and H. Rosenberg.

Lemma 2.4.

Let $\Sigma$ be an immersed orientable hypersurface in ${\mathbb{M}}^m \times {\mathbb{R}}$ (with or without boundary) and $N$ be its normal unit vector field. Then

$$L_{r}(h)=(r+1)S_{r+1}n ,$$

for $0\leqslant r \leqslant m,$ where $h$ denotes the height function of $\Sigma ,$ and $n=\langle \frac{\partial }{\partial t},N\rangle .$

Lemma 2.5.

Let $\Sigma$ be an immersed hypesurface in ${\mathbb{M}}^{m} \times {\mathbb{R}}.$ Then we have

$$L_{r}(n)=-n(S_1S_{r+1}-(r+2)S_{r+2} +tr(T_{r} \overline{R}_N))-E^T_0(S_{r+1}).$$

Proof.

The proof uses the same argument as the proof of Lemma 4.2 in 4, with the only difference being that in our case $S_r$ is not assumed to be constant on $\Sigma .$

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Remark 2.6.

If either $r=0$ or $r=1,$ we get respectively for $m=2$ the following formulae:

$$L_{0}(n)=\Delta n=-n(S_1^2-2S_{2} +tr(T_{0} \overline{R}_N))-E^T_0(S_1),$$

$$L_{1}(n)=-n(S_1S_2 +tr(T_{1} \overline{R}_N))-E^T_0(S_2).$$

If the manifold ${\mathbb{M}}^{m}$ has constant sectional curvature, then we are able to express all terms of $L_{r-1}(n)$ given by Lemma 2.5 in terms of the curvatures $S_r$.

We denote by $X^h$ the horizontal component of $X \in T_p ({\mathbb{M}}^m(c) \times {\mathbb{R}}),$ by $e_i$ the principal directions of $A$ and by $A_i$ the restriction of $A$ to the $(m-1)$-dimensional space normal to $e_i.$

Lemma 2.7.

Let $\Sigma$ denote a hypersurface immersed in ${\mathbb{M}}^m(c) \times {\mathbb{R}}.$ For $0 \leqslant r \leqslant m,$ the following holds:

$$tr(T_{r} \overline{R}_N)=c(m-r)S_{r}.$$

Proof.

$$tr(T_{r} \overline{R}_N)(p)= \sum _i \langle e_i,T_{r+1}\overline{R}_N(e_i)\rangle (p)=\sum _i S_{r+1}(A_i)\overline{R} (N,e_i,N,e_i)(p)$$

$$=\sum _i S_{r+1}(A_i) K(e_i^h,N^h)|e_i^h \wedge N^h|^2(p)=c \sum _i S_{r+1}(A_i)=c(m-r)S_{r}.$$ ■

Remark 2.8.

If either $r=0$ or $r=1,$ we get respectively for $m=2$ the following formulae:

$$tr(T_{0} \overline{R}_N)=2 c S_0=2c,$$

$$tr(T_{1} \overline{R}_N)= c S_1=2cH_1.$$

In the next section we will use the results presented here to show that the special Weingarten surfaces of elliptic type satisfy an interior and a boundary maximum principle and a height estimate under additional conditions.

2.2. Maximum principle for special Weingarten surfaces

Let $\Sigma$ be an oriented connected hypersurface immersed in ${\mathbb{M}}^m(c) \times {\mathbb{R}}$ and $f \in {\mathcal{C}}^1([0,\infty )).$ Let us suppose that the first and second mean curvatures $H_1(s),H_2(s)$ of $\phi _s(\Sigma )$ (see Definition 2.1) satisfy

$$\begin{equation} H_1-f(H_1^2-H_2)=0. {\tag{4}} \end{equation}$$

The first variation of the left member of this identity at $s=0$ gives us

$$\begin{equation} \left((1-2H_1f'(H_1^2-H_2)) \frac{\partial H_1}{\partial s}+ f'(H_1^2-H_2)\frac{\partial H_2}{\partial s}\right)_{|_{\{s=0\}}}=0. {\tag{5}} \end{equation}$$

From 3, the principal parts of $\partial _s H_1(0)=\frac{1}{m} \partial _s S_1(0)$ and $\partial _s H_2(0)= \frac{2}{m(m-1)} \partial _s S_2(0)$ are respectively $L_0/m$ and $\frac{2}{m(m-1)} L_1.$

When $m=2$ the linearized operator of 4 reduces to

$$L_f=\left( \frac{1-2ff'}{2} \right)\Delta +f' L_1.$$

As in 14, page 294, we can prove the following lemma.

Lemma 2.9.

If the function $f$ is elliptic, that is, $4x(f'(x))^2<1$ for all $x \geq 0,$ then the eigenvalues of the operator $L_f$ are positive. In other terms $L_f$ is elliptic.

Remark 2.10.

Lemma 2.9 says that $H_1=f(H_1^2-H_2)$ is elliptic in the sense of Hopf and the solutions of this equation satisfy an interior and a boundary maximum principle (see 7, pages 156-158).

Let $\Sigma _1,\Sigma _2$ be two oriented special Weingarten surfaces in ${\mathbb{M}}^2(c) \times {\mathbb{R}}$ satisfying 1 for the same function $f$, whose unit normal vectors coincide at a common point $p$. For $i=1,2$, we can write $\Sigma _i$ locally around $p$ as a graph of a function $u_i$ over a domain in $T_p\Sigma _1=T_p\Sigma _2$ (in exponential coordinates). We will say that $\Sigma _1$ is above $\Sigma _2$ in a neighbourhood of $p$, and we will write $\Sigma _1\geq \Sigma _2$ if $u_1\geq u_2$.

Proposition 2.11 (Maximum Principle 7).

Let $\Sigma _1,\Sigma _2$ be two special Weingarten surfaces in ${\mathbb{M}}^2(c) \times {\mathbb{R}}$ with respect to the same elliptic function $f$. Let us suppose that

•

$\Sigma _1$ and $\Sigma _2$ are tangent at an interior point $p\in \Sigma _1\cap \Sigma _2$ or

•

there exists $p\in \partial \Sigma _1\cap \partial \Sigma _2$ such that both $T_p\Sigma _1=T_p\Sigma _2$ and $T_p\partial \Sigma _1=T_p\partial \Sigma _2$.

Also suppose that the unit normal vectors of $\Sigma _1,\Sigma _2$ coincide at $p$. If $\Sigma _1\geq \Sigma _2$ in a neighbourhood $U$ of $p$, then $\Sigma _1=\Sigma _2$ in $U$. In the case $\Sigma _1,\Sigma _2$ have no boundary, $\Sigma _1=\Sigma _2$.

To show the main theorem we need the following result.

Lemma 2.12.

If $H_1(s)$ and $H_2(s)$ verify $H_1-f(H_1^2-H_2)=0$ for each $s,$ then

$$(1-2ff')E_s^T(H_1)+f'E_s^T(H_2)=0.$$

Proof.

It is sufficient to find the explicit expression of $E_s^T(H_1-f(H_1^2-H_2))$ and to use the fact that it vanishes. It holds that

$$E_s^T(H_1)-E_s^Tf(H_1^2-H_2)=E_s^T(H_1)-2ff'E_s^T(H_1)+ f'E_s^T(H_2)=0.$$

That is, $(1-2ff')E_s^T(H_1)+f'E_s^T(H_2)=0.$

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Now we give the proof of Theorem 1.1.

Proof.

Let $m$ denote the maximum of $h$ on $\Sigma .$ It is sufficient to give the proof assuming that $\Sigma$ is a graph on the slice $\{t=0\},$ where $t$ denotes the coordinate on ${\mathbb{R}}.$ Indeed by coming from infinity with horizontal slices and applying the Alexandrov reflection to $\Sigma ,$ we see that the part of $\Sigma$ above the plane $t=m/2$ is a graph over a domain in this plane. The estimate we are going to prove is $|h|\leqslant 1/l$ when $\Sigma$ is a graph. We can assume $h\geqslant 0$ on $\Sigma$; otherwise we apply the following argument to the part of $\Sigma$ above $\{t=0\}$ and to the part of $\Sigma$ below $\{t=0\}.$ We orient the unit normal vector to $\Sigma$ so that $n\leqslant 0.$ We set $\varphi =lh+n.$ On $\partial \Sigma$ we have $\varphi =n\leqslant 0.$ If we show that $L_f \varphi \geqslant 0$ on $\Sigma ,$ from ellipticity of $L_f$ and the maximum principle we get that $\varphi \leqslant 0$ on $\Sigma ,$ that is, $h\leqslant -n/l\leqslant 1/l.$ By Lemma 2.4, Remarks 2.6 and 2.8, and Lemma 2.12 we get

$$L_f(\varphi )=L_f(lh+n)=lL_f(h)+L_f(n)$$

$$=l\left( \left( \frac{1-2ff'}{2} \right)\Delta h +f' L_1 h \right)+ \left( \frac{1-2ff'}{2} \right)\Delta n +f' L_1 n$$

$$=l\left( \left( \frac{1-2ff'}{2} \right) 2 H_1 n + 2f'H_2n \right)$$

$$-\left( \frac{1-2ff'}{2}\right)((4H_1^2-2H_2+2c)n+2E_0^T(H_1)) -f'((2H_1H_2+2cH_1)n+ E_0^T(H_2))$$

$$=-n\left( \left( 1-2ff' \right) \left( 2H_1^2 -H_2 +c \right)+ f'\left( 2H_1H_2+2cH_1 \right) \right)$$

$$+\ l\left( \left( 1-2ff'\right) H_1 n + 2f'H_2n \right) -\left( 1-2ff' \right)E_0^T(H_1)+f' E_0^T(H_2)$$

$$=-n\left( \left( 1-2ff' \right) \left( 2H_1^2 -H_2 +c -lH_1\right)+ f'\left( 2H_1H_2+2cH_1 -2lH_2 \right) \right).$$

We replace $H_1$ by $f$ and $H_1^2-H_2$ by $x.$ We get

$$-n\left( \left( 1-2ff' \right) \left( f^2+x +c-lf \right)+ f'\left( 2f (f^2-x)+2cf -2l(f^2-x) \right) \right)$$

$$=-n\left(f^2-lf+x+2xlf'-4xff' +c \right).$$ Such a quantity can be written as

$$-n \left(f(f-l)(1-4xf'^2)+c + x(1-2ff')^2+2xlf'(1-2ff') \right),$$

which is non-negative if

$$f(f-l)(1-4xf'^2)+c + x(1-2ff')^2+2xlf'(1-2ff')$$

$$= f^2+c+x(1-4ff') -l[f(1-4xf'^2)-2xf'(1-2ff')]$$

$$=f^2+c+x(1-4ff') -l[f-2xf']\geqslant 0.$$

Suppose that

$$f-2xf'>0.$$

Then

$$l \leqslant \frac{f^2+c+x(1-4ff')}{f-2xf'}= f+\frac{c+x(1-2ff')}{f-2xf'}.$$

As $x=f^2-K_e$ we have

$$l \leqslant f+\frac{c-K_e +f^2 -2(f^2-K_e)ff'}{f-2(f^2-K_e)f'}= 2f+ \frac{c-K_e}{f-2(f^2-K_e)f'}=:L.$$

If we assume $f^2+c+x(1-4ff')>0,$ then $L >0.$ We conclude that the best constant $l$ is

$$l = \min _\Sigma \,\left( 2f+ \frac{c-K_e}{f-2(f^2-K_e)f'} \right)= \min _\Sigma \,\left( 2f+ \frac{c-K_e}{f(1-2ff')+2K_e f'} \right).$$ ■

Now we give the proof of Corollary 1.2.

Proof.

If $f'=0,$ that is, $\Sigma$ has constant mean curvature, we get the estimate

$$h \leqslant \frac{1}{l}= \max _\Sigma \frac{H}{2H^2+c-K_e}= \frac{H}{2H^2+c-\max _\Sigma K_e}$$

under the assumption $2H^2+c-\max _\Sigma K_e >0.$

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Remark 2.13.

If $c<0$, then the estimate of Corollary 1.2 holds if $H > \sqrt \frac{\max _\Sigma K_e-c}{2}.$ We observe that $\sqrt \frac{\max _\Sigma K_e-c}{2} \geqslant \sqrt \frac{-c}{2}.$ As a consequence such an estimate is not sharp because the smallest value of the mean curvature of a compact surface in ${\mathbb{H}}^2(c) \times {\mathbb{R}}$ equals $\sqrt {-c}/2.$ Optimal estimates have been derived in 1. If $c=0$ and $\Sigma$ is a semisphere (in particular $H^2=K_e$ holds), we get the well known estimate $|h| \leqslant 1/H.$

2.3. Proof of Theorem 1.4

Proof.

Let us suppose by absurdity that $S$ is a properly embedded SWET surface with finite topology and one end with respect to a function $f$ which also satisfies the hypotheses of Theorems 1.1, 1.3.

Let us denote by $D$ a constant bigger than $\max \{C,d \},$ where $C$ is the bound for the horizontal height given by Theorem 1.3 and $d$ is the horizontal diameter of the sphere of constant mean curvature equal to $H_0$.

Let $p$ denote a point in $S$ and $\gamma$ a horizontal geodesic containing $p$. Let $p_1,p_2$ be two points in $\gamma$ such that $dist_{{\mathbb{H}}^2(c)}(p,p_1)=dist_{{\mathbb{H}}^2(c)}(p_1,p_2)=D,$ $dist_{{\mathbb{H}}^2(c)}(p,p_2)=2D.$ Let $P_1,P_2$ be two vertical totally geodesic planes intersecting orthogonally with $\gamma$ at $p_1,p_2$ respectively.

We now use the following variant of the Plane Separation Lemma proved in 13. Its proof is exactly the same.

Lemma 2.14.

Let $S$ be a properly embedded SWET annulus in ${\mathbb{H}}^2(c) \times {\mathbb{R}}.$ Suppose that $f \geq H_0 > \frac{\sqrt {-c}}{2}.$ Let $P_1$ and $P_2$ be two vertical totally geodesic planes. Assume that the distance between $P_1$ and $P_2$ is bigger than the horizontal diameter of the sphere of constant mean curvature $H_0.$ Denote by $P_1^+$ and $P_2^+$ the components of ${\mathbb{H}}^2(c) \times {\mathbb{R}}\setminus P_j$ such that $P_1^+ \cap P_2^+ =\emptyset$. Then all the connected components of $S \cap P_1^+$ or $S \cap P_2^+$ are compact.

The distance between $P_1$ and $P_2$ equals $D>d$; then the previous lemma applies. Suppose that all of the connected components of $S \cap P_1^+$ are compact. By construction the plane $P_1$ is at distance $D$ to the point $p \in S \cap P_1^+.$ As $D>C,$ $C$ being the bound for the horizontal distance, this would contradict Theorem 1.3. Then all the connected components of $S \cap P_2^+$ are compact. By Theorem 1.3 the points of $S \cap P_2^+$ are at a distance to $P_2$ which has to be smaller than $C.$

If we use the same argument after replacing $\gamma$ by every other horizontal geodesic line passing by $p$, we can prove that $S$ is located at finite distance to $\{p\} \times {\mathbb{R}}.$ In other words, $S$ is contained in a vertical cylinder.

As by hypothesis $S$ has exactly one end, we can assume that $S$ is contained in the halfspace $\{t \leq 0\}$ and is tangent to $\{t=0 \}.$

For $z<0$ we consider the reflection in $\{t=z \}$ of the compact piece $S_z$ of $S$ contained in $\{z \leq t \leq 0\}.$ We will show that $S_z$ is a vertical graph on $\{t=z \}$ for any $z<0.$ We denote by $S_z^R$ the reflection of $S_z$ in $\{t=z \}.$ We observe that $S_z^R$ does not have common tangent points with $S$. Otherwise by the Maximum Principle, Proposition 2.11, we could conclude that $S$ is compact. For the same reason $S_z^R$ and $S_z$ are not orthogonal to $\{t=z \}$ for any $z<0.$ This proves that $S_z$ is a graph.

Now we can choose $z_0<0$ with $|z_0|$ big enough so that the height of the compact graph $S_{z_0}$ (having boundary on the plane $\{t =z_0\}$) is arbitrarily big. This contradicts Theorem 1.1.

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