Height estimates and half-space theorems for hypersurfaces in product spaces of the type $ℝ\times M^n$

Abstract

We obtain height estimates and half-space theorems concerning a wide class of hypersurfaces immersed into a product space $ℝ\times M^n$, the so-called generalized linear Weingarten hypersurfaces, which extends that one having some constant higher order mean curvature.

Key words
Product spaces; generalized linear Weingarten hypersurfaces; height estimates; half-space theorems

1 - INTRODUCTION

The last decades have seen a steadily growing interest in the study of a priori estimates for the height function of constant mean curvature compact graphs or, more generally, compact hypersurfaces with boundary having some constant higher order mean curvature. This problem has gained special attention, being considered by several authors probably motivated by the fact that these estimates turn out to be a very useful tool in order to investigate existence and uniqueness results for complete hypersurfaces with constant higher order mean curvature, as well as to obtain information on the topology at infinity of such hypersurfaces (see, for instance, Aledo et al. 2008, 2010, Alías & Dajczer 2007, Cheng & Rosenberg 2005, Espinar et al. 2009, Heinz 1969, Hoffman et al. 2006, Korevaar et al. 1992, 1989, Rosenberg 1993).

A height estimate of compact graphs with positive constant mean curvature in the Euclidean space ${ℝ}^{n+1}$ and boundary in a hyperplane, were first obtained by Heinz 1969. More specifically, denoting by $H$ the mean curvature, Heinz proved that the height of such a graph can rise at most $1/H$. More than twenty years after that, Korevaar et al. 1992 obtained a sharp bound for compact graphs and for compact embedded hypersurfaces in the hyperbolic space ${ℍ}^{n+1}$ with nonzero constant mean curvature and boundary in a horosphere. More generally, given an arbitrary Riemannian manifold $M^n$, height estimates in the product space $ℝ\times M^n$ for constant mean curvature compact embedded hypersurfaces with boundary in a slice were exhibited by Hoffman et al. 2006 and Aledo et al. 2008, for $n=2$, and by Alías & Dajczer 2007, for an arbitrary dimension $n$.

Regarding hypersurfaces having some constant higher order mean curvature, this was done firstly by Rosenberg 1993, who proved height estimates for compact embedded hypersurfaces with zero boundary values either in the Euclidean space or in the hyperbolic space, generalizing the previous estimates of Heinz and Korevaar. Later on, Cheng & Rosenberg 2005 were able to generalize these estimates for compact graphs with some constant higher order mean curvature in the product manifold $ℝ\times M^n$, with boundary in a slice. As application of their height estimates, they used the Alexandrov’s reflection technique to prove that a noncompact properly embedded hypersurface having constant higher order mean curvature in $ℝ\times M^n$, where $M^n$ is a compact manifold with sectional curvature bounded from below, has at least two ends or, equivalently, it cannot lie in a half-space. The same technique was used by Hoffman et al. 2006 in order to obtain some information on the topology at infinity of properly embedded surfaces of constant mean curvature in $ℝ\times M^2$. More recently, Rosenberg et al. 2013 showed that an entire minimal graph with nonnegative height function in a product space $ℝ\times M^n$, whose base $M^n$ is a complete Riemannian manifold having non-negative Ricci curvature and with sectional curvature bounded from below, must be a slice.

Proceeding with the picture described above, in this paper our aim is to obtain height estimates and half-space theorems of a wide class of hypersurfaces immersed into a product space $ℝ\times M^n$, which extends that one having some constant higher order mean curvature. Precisely, we consider in $ℝ\times M^n$ generalized linear Weingarten hypersurfaces, by meaning that there exists a linear relation involving some of the corresponding higher order mean curvatures (for more details, see Section 3). We point out that our results offer improvements of those ones obtained in Alías & Dajczer 2007, Alías et al. 2016, Cheng & Rosenberg 2005 and Hoffman et al. 2006. Furthermore, we are able to prove half-space theorems for complete noncompact generalized linear Weingarten hypersurfaces in $ℝ\times M^n$, generalizing some results of Cheng & Rosenberg 2005 and Hoffman et al. 2006. Recently, the authors proved similar results for the case of hypersurfaces immersed into warped product manifolds (see de Lima & de Lima 2018). However, as we will see, the results presented there do not contemplate those obtained here.

This manuscript is organized in the following way: In Section 2 we introduce some basic facts and notations that will appear in the proofs of our results. In particular, we recall some geometric conditions which guarantee the ellipticity of the linearized operator of the higher order mean curvature (see Lemmas 2 and 3). In Section 3, we establish our first main results concerning height estimates of compact generalized linear Weingarten hypersurfaces in $ℝ\times M^n$ (see Theorems 1 and 2). In Section 4, as application of our height estimates, we prove half-space theorems related to noncompact generalized linear Weingarten hypersurface immersed in $ℝ\times M^n$, supposing that the fiber $M^n$ is compact (see Theorems 3 and 4). Finally, when $M^n$ is not necessarily compact, using a generalized version of the Omori-Yau maximum principle for trace type differential operators, we prove other half-space theorem, which is of independent interest by itself (see Theorem 5).

2 - PRELIMINARIES

In this section we will introduce some basic facts and notations that will appear along the paper. In this sense, along this work we will always consider $M^n$ a (connected) $n$-dimensional Riemannian manifold and $I\subset ℝ$ an open interval in $ℝ$. Let us denote by ${\overline{M}}^{n+1}=I\times M^n$ the product manifold endowed with the Riemannian metric

$$⟨,⟩={\pi }_I^{*}\left(d{t}^2\right)+{\pi }_M^{*}(⟨,{⟩}_M),$$

where ${\pi }_I$ and ${\pi }_M$ denote the canonical projections from $I\times M^n$ onto each factor, $⟨,{⟩}_M$ is the Riemannian metric on the fiber $M^n$ and $I$ is endowed with the metric $d{t}^2$. Observe that ${\partial }_t$ is a unitary vector field globally defined on ${\overline{M}}^{n+1}=I\times M^n$, which determines on ${\overline{M}}^{n+1}$ a codimension one foliation by totally geodesic slices $\{t\}\times M$.

Throughout this paper, we will study (connected) two-sided hypersurfaces $\psi :{\Sigma }^n\to {\overline{M}}^{n+1}$ immersed into the product Riemannian manifold ${\overline{M}}^{n+1}=I\times M^n$, which means that there exists a unitary normal vector field $N$ globally defined on ${\Sigma }^n$. As usual, we also denote by $⟨,⟩$ the metric of ${\Sigma }^n$ induced via $\psi$. In this setting, we consider two particular functions naturally attached to the two-sided hypersurface ${\Sigma }^n$, namely, the (vertical) height function $h={\pi }_{ℝ}\circ \psi$ and the angle function $\Theta =⟨N,{\partial }_t⟩$.

Let us denote by $A:𝔛\left(\Sigma \right)\to 𝔛\left(\Sigma \right)$ the shape operator (or Weingarten endomorphism) of ${\Sigma }^n$ in ${\overline{M}}^{n+1}=I\times M^n$ with respect to $N$, which is given by $AX=-{\overline{\nabla }}_{X}N$, where $\overline{\nabla }$ stands for the Levi-Civita connection of ${\overline{M}}^{n+1}$. A fact well known is that the curvature tensor $R$ of the hypersurface ${\Sigma }^n$ can be described in terms the shape operator $A$ and of the curvature tensor $\overline{R}$ of the ambient space ${\overline{M}}^{n+1}=I\times M^n$ by the Gauss equation given by

$$R(X,Y)Z={\left(\overline{R}(X,Y)Z\right)}^{\top }+⟨AX,Z⟩AY-⟨AY,Z⟩AX$$(2.1)

for every tangent vector fields $X,Y,Z\in 𝔛\left(\Sigma \right)$, where ${\left(\right)}^{\top }$ denotes the tangential component of a vector field in $𝔛\left({\overline{M}}^{n+1}\right)$ along ${\Sigma }^n$.

Associated with the shape operator $A$ there are $n$ algebraic invariants, which are the elementary symmetric functions $S_r$ of its principal curvatures ${\kappa }_1,\dots ,{\kappa }_n$, given by

$$S_r=S_r({\kappa }_1,\dots ,{\kappa }_n)=\sum _{i_1\dots i_r}{\kappa }_{i_1}\cdots {\kappa }_{i_r},1\le r\le n.$$

As it is well known, the $r$-mean curvature $H_r$ of the hypersurface ${\Sigma }^n$ is defined by

$$\left(\genfrac{}{}{0ex}{}{n}{r}\right)H_r=S_r({\kappa }_1,\dots ,{\kappa }_n).$$

In particular, when $r=1$,

$$H_1=\frac{1}{n}\sum _i{\kappa }_i=\frac{1}{n}\mathrm{\text{tr}}\left(A\right)=H$$

is just the mean curvature of ${\Sigma }^n$. When $r=2$, $H_2$ defines a geometric quantity which is related to the (intrinsic) scalar curvature $S$ of the hypersurface. For instance, when the ambient space has constant sectional curvature $c$, it follows from the Gauss equation that $S=(n-1)(c+H_2)$. In general, it also follows from Gauss equation of the hypersurface that when $r$ is odd $H_r$ is extrinsic (and its sign depends on the chosen orientation), while when $r$ is even $H_r$ is an intrinsic geometric quantity.

It is a classical fact that the higher order mean curvatures satisfy a very useful set of inequalities, usually alluded as Newton’s inequalities. For future reference, we collect them here. A proof can be found in Hardy et al. 1989.

Lemma 1. Let $\psi :{\Sigma }^n\to {\overline{M}}^{n+1}$ be a two-sided hypersurface immersed into a product space ${\overline{M}}^{n+1}=I\times M^n$. For each $1\le r\le n$, if $H_1,\dots ,H_r$ are nonnegative on ${\Sigma }^n$, then:

• (a) $H_r{H}_{r+2}\le H_{r+1}^2$;

• (b) $H_1\ge H_2^{1/2}\ge \dots \ge H_r^{1/r}$,

and equality holds only at umbilical points.

For each $0\le r\le n$, one defines the $r$-th Newton transformation $P_r:𝔛\left(\Sigma \right)\to 𝔛\left(\Sigma \right)$ of the hypersurface ${\Sigma }^n$ by setting $P_0=I$ (the identity operator) and, for $1\le r\le n$, via the recurrence relation

$$P_r=\left(\genfrac{}{}{0ex}{}{n}{r}\right)H_{r}I-A{P}_{r-1},$$

Equivalently,

$$P_r=\sum _{j=0}^r\left(\genfrac{}{}{0ex}{}{n}{j}\right){(-1)}^{r-j}{H}_j{A}^{r-j},$$

so that the Cayley-Hamilton theorem gives $P_n=0$. Observe also that when $r$ is even, the definition of $P_r$ does not depend on the chosen unitary normal vector field $N$, but when $r$ is odd there is a change of sign in the definition of $P_r$. Moreover, it is easy to see that each $P_r$ is a self-adjoint operator which commutes with shape operator $A$, that is, if a local orthonormal frame on ${\Sigma }^n$ diagonalizes $A$, then it also diagonalizes each $P_r$. More specifically, if $\{E_1,\dots ,E_n\}$ is such a local orthonormal frame with $A\left(E_i\right)={\kappa }_i{E}_i$, then

$$P_r\left(E_i\right)={\mu }_{i,r}{E}_i,$$

where

$${\mu }_{i,r}=\sum _{i_1\cdots i_r,i_j\ne i}{\kappa }_{i_1}\cdots {\kappa }_{i_r}.$$

It follows from here that for each $0\le r\le n-1$, we have

$$\mathrm{\text{tr}}\left(P_r\right)=c_r{H}_r,\mathrm{\text{with}}{c}_r=(n-r)\left(\genfrac{}{}{0ex}{}{n}{r}\right)=(r+1)\left(\genfrac{}{}{0ex}{}{n}{r+1}\right).$$

Let $\nabla$ stand for the Levi-Civita connection of the two-sided hypersurface ${\Sigma }^n$. Associated to each Newton transformation $P_r$, one has the second order linear differential operator $L_r:C^{\infty }\left(\Sigma \right)\to C^{\infty }\left(\Sigma \right)$ for $r=0,1,\dots ,n-1$, defined by

$$L_{r}u=\mathrm{\text{tr}}(P_r\circ \mathrm{\text{hess}}u),$$

where $\mathrm{\text{hess}}u:𝔛\left(\Sigma \right)\to 𝔛\left(\Sigma \right)$ denotes the self-adjoint linear operator metrically equivalent to the Hessian of $u$, $\mathrm{\text{Hess}}u$, which are given by

$$\mathrm{\text{hess}}u\left(X\right)={\nabla }_X\nabla u\mathrm{\text{and}}\mathrm{\text{Hess}}(X,Y)=⟨\mathrm{\text{hess}}u\left(X\right),Y⟩,$$

respectively, for all $X,Y\in 𝔛\left(\Sigma \right)$. In particular, $L_0=\Delta$, the Laplacian of ${\Sigma }^n$, which is always an elliptic operator in divergence form. More generally, it is well known that the operator $L_r$ is elliptic if and only if $P_r$ is positive definite.

For our applications, it will be useful to have some geometric conditions which guarantee the ellipticity of $L_r$ when $r\ge 1$. For $r=1$, the next lemma assures the ellipticity of $L_1$ (see Lemma 3.10 of Elbert 2002).

Lemma 2. Let $\psi :{\Sigma }^n\to {\overline{M}}^{n+1}$ be a two-sided hypersurface immersed into a product space ${\overline{M}}^{n+1}=I\times M^n$. If $H_{2}0$ on $\Sigma$, then $L_1$ is elliptic or, equivalently, $P_1$ is positive definite (for an appropriate choice of the orientation $N$).

When $r\ge 2$, the following lemma give us sufficient conditions to guarantee the ellipticity of $L_r$. The proof is given in Proposition 3.2 of Cheng & Rosenberg 2005 (see also Proposition 3.2 of Barbosa & Colares 1997).

Lemma 3. Let $\psi :{\Sigma }^n\to {\overline{M}}^{n+1}$ be a two-sided hypersurface (with or without boundary) immersed into a product space ${\overline{M}}^{n+1}=I\times M^n$ with $H_{r+1}0$ on ${\Sigma }^n$, for some $2\le r\le n-1$. If there exists an interior point $p$ of ${\Sigma }^n$ such that all the principal curvatures at $p$ are nonnegative, then for all $1\le k\le r$ the operator $L_k$ is elliptic and the $(k+1)$-mean curvature $H_{k+1}$ is positive.

Next, we close this section with the following formulas, which will be essential for the proofs of our main results (for more details see Proposition 6 and Lemma 27 of Alías et al. 2013).

Proposition 1. Let $\psi :{\Sigma }^n\to {\overline{M}}^{n+1}$ be a two-sided hypersurface immersed into a product space ${\overline{M}}^{n+1}=I\times M^n$. For every $r=0,\dots ,n-1$:

• (a) The height function satisfies

  $$\mathrm{\text{Hess}}h(X,Y)=\Theta ⟨AX,Y⟩\mathrm{\text{and}}{L}_{r}h=c_r\Theta H_{r+1},$$
  where$c_r:=(n-r)\left(\genfrac{}{}{0ex}{}{n}{r}\right)=(r+1)\left(\genfrac{}{}{0ex}{}{n}{r+1}\right)$.

• (b) The angle function satisfies

  $$\begin{array}{ccc}\hfill L_r\Theta & =\hfill & \hfill -\frac{c_r}{r+1}⟨\nabla H_{r+1},\nabla h⟩-\frac{c_r\Theta }{r+1}(n{H}_1{H}_{r+1}-(n-r-1)H_{r+2})\\ \hfill & -\hfill & \hfill \Theta \sum _{i=1}^n{\mu }_{i,r}{K}_M(N^{*},E_i^{*}){|N^{*}\wedge E_i^{*}|}^2,\end{array}$$
  where$\{E_1,\dots ,E_n\}$is an orthonormal frame on${\Sigma }^n$diagonalizing$A$, $K_M$denotes the sectional curvature of the fiber$M^n$, ${\mu }_{i,r}$stands for the eigenvalues of$P_k$and, for every vector field$X\in 𝔛\left(\overline{M}\right)$, $X^{*}$is the orthogonal projection on$TM$.

3 - HEIGHT ESTIMATES OF GENERALIZED LINEAR WEINGARTEN HYPERSURFARCES

This section is devoted to establish our results concerning to estimates of the height function $h$ of a wide class of two-sided hypersurfaces immersed into a product Riemannian manifold ${\overline{M}}^{n+1}=ℝ\times M^n$, which extends that one having some constant higher order mean curvature. Specifically, let us consider $\psi :{\Sigma }^n\to {\overline{M}}^{n+1}$ a two-sided hypersurface immersed into a product space ${\overline{M}}^{n+1}=ℝ\times M^n$. We say that ${\Sigma }^n$ is $(r,s)$-linear Weingarten, for some $0\le r\le s\le n-1$, if there exist nonnegative real numbers $b_r,\dots ,b_s$ (at least one of them nonzero) such that the following linear relation holds on ${\Sigma }^n$:

$$\sum _{k=r}^s{b}_k{H}_{k+1}=d\in ℝ.$$(3.1)

Thus, naturally attached to a $(r,s)$-linear Weingarten two-sided hypersurface we have the constant $d$ given by (3.1). We note that the $(r,r)$-linear Weingarten two-sided hypersurfaces are exactly the two-sided hypersurfaces having $d=H_{r+1}$ constant. On the other hand, if the ambient space has zero sectional curvature and taking into account that in this case $\overline{S}=H_2$, where $\overline{S}$ stands for the normalized scalar curvature of ${\Sigma }^n$, we observe that the $(0,1)$-linear Weingarten two-sided hypersurfaces are called simply linear Weingarten two-sided hypersurfaces. Throughout this paper, we will always denote by $d$ the constant given by equation (3.1).

Now, we are in position to state and prove our first main result. More precisely, we will establish an estimate for the height function concerning $(r,s)$-linear Weingarten two-sided hypersurface in a product space $ℝ\times M^n$.

Theorem 1. Let ${\overline{M}}^{n+1}=ℝ\times M^n$ be a product space whose the fiber $M^n$ has nonnegative sectional curvature $K_M$. Let $\psi :{\Sigma }^n\to {\overline{M}}^{n+1}$ be a compact $(r,s)$-linear Weingarten two-sided hypersurface with $(s+1)$-mean curvature $H_{s+1}\ne 0$ on ${\Sigma }^n$, for some $0\le s\le n-1$, and boundary $\partial {\Sigma }^n$ contained into the slice $\{t_0\}\times M^n$ for some $t_0\in ℝ$. Suppose that the angle function $\Theta$ does not change sign on ${\Sigma }^n$. Then,

• (a) Either $maxh\ne t_0$ and

  $${\Sigma }^n\subset [t_0,t_0+\frac{1}{min\left|H_1\right|}]\times M^n,$$

• (b) or $minh\ne t_0$ and

  $${\Sigma }^n\subset [t_0-\frac{1}{min\left|H_1\right|},t_0]\times M^n.$$

Proof. First of all it is clear from our hypothesis on the $(s+1)$-mean curvature that either $maxh\ne t_0$ or $minh\ne t_0$. So, we begin by assuming that $maxh\ne t_0$ and let us choose an interior point $p_0$ of ${\Sigma }^n$ such that the height function reaches its maximum and the orientation so that $\Theta \le 0$. Then, Proposition 1 yields

$$0\ge \mathrm{\text{Hess}}h\left(p_0\right)(v,v)=\Theta \left(p_0\right)⟨Av,v⟩\left(p_0\right),\forall v\in T_{p_0}\Sigma ,$$

that is, at $p_0$ all the principal curvatures are nonnegative. Since we are assume that $H_{s+1}\ne 0$ on ${\Sigma }^n$, we must have $H_{s+1}0$ on ${\Sigma }^n$. In particular, we can apply Lemma 3 (or Lemma 2 if $s=1$) to guarantee the ellipticity of the operator $L_k$ for every $k=r,\dots ,s$ and $H_{k+1}$ is positive on ${\Sigma }^n$ for every $0\le k\le s$. So, for instance, we have

$$L_{s}h=c_s\Theta H_{s+1}\le 0$$

and, consequently, by the weak maximum principle we obtain that $h\ge t_0$ on ${\Sigma }^n$.

Now let us consider on ${\Sigma }^n$ the smooth function $\phi =ch+\Theta$, where $c\in ℝ$ is a positive constant to be chosen in an appropriate way. Then Proposition 1 gives

$$\begin{array}{ccc}\hfill L_k\phi & =\hfill & \hfill -\frac{c_k}{k+1}⟨\nabla H_{k+1},\nabla h⟩-\Theta \left(\genfrac{}{}{0ex}{}{n}{k+1}\right)(n{H}_1{H}_{k+1}-(n-k-1)H_{k+2}\\ \hfill & \hfill & \hfill -(k+1)c{H}_{k+1})-\Theta \sum _{i=1}^n{\mu }_{i,k}{K}_M(N^{*},E_i^{*}){|N^{*}\wedge E_i^{*}|}^2,\end{array}$$(3.2)

where $\{E_1,\dots ,E_n\}$ is an orthonormal frame on ${\Sigma }^n$ diagonalizing $A$ with $P_k{E}_i={\mu }_{i,k}{E}_i$, for every $i=1,\dots ,n$ and $k=r,\dots ,s$, and $X^{*}$ denotes the orthogonal projection on $TM$.

Since $H_{k+1}$ is positive for every $k=0,\dots ,s$, from Lemma 1 we get

$$H_1{H}_{k+1}-H_{k+2}\ge H_1{H}_{k+1}-H_{k+1}^2{H}_k^{-1}={\displaystyle \frac{H_{k+1}}{H_k}}(H_1{H}_k-H_{k+1}).$$

By using once more Lemma 1 it follows from here that

$$H_1{H}_{k+1}-H_{k+2}\ge {\displaystyle \frac{H_{k+1}}{H_k}}(H_1{H}_k-H_k^{(k+1)/k})=H_{k+1}(H_1-H_k^{1/k})\ge 0.$$

Then, the previous inequality implies that

$$\begin{array}{ccc}\hfill n{H}_1{H}_{k+1}-(n-k-1)H_{k+2}-(k+1)c{H}_{k+1}& =\hfill & \hfill (k+1)H_{k+1}(H_1-c)\\ \hfill & +\hfill & \hfill (n-k-1)(H_1{H}_{k+1}-H_{k+2})\\ \hfill & \ge \hfill & \hfill (k+1)H_{k+1}(H_1-c)\ge 0,\end{array}$$(3.3)

provided that $c:=min{H}_1$.

On the other hand, since the operator $L_k$ is elliptic for every $k=r,\dots ,s$ or, equivalently, $P_k$ is positive definite, we get that its eigenvalues ${\mu }_{i,k}$ are all positive on ${\Sigma }^n$. Then, by our assumption on the sectional curvature $K_M$ of the fiber $M^n$ we have

$$\begin{array}{c}\hfill \sum _{i=1}^n{\mu }_{i,k}{K}_M(N^{*},E_i^{*}){|N^{*}\wedge E_i^{*}|}^2\ge 0,\end{array}$$

Hence, by using (3.3) and taking into account that $\Theta \le 0$, we infer from (3.2) and the previous inequality that

$$L_k\phi \ge -\frac{c_k}{k+1}⟨\nabla H_{k+1},\nabla h⟩.$$(3.4)

Proceeding, we introduce the following second order linear differential operator $L:C^{\infty }\left(\Sigma \right)\to C^{\infty }\left(\Sigma \right)$ defined by

$$\begin{array}{ccc}\hfill L& =\hfill & \hfill \sum _{k=r}^s(k+1)c_k^{-1}{b}_k{L}_k\\ \hfill & =\hfill & \hfill \mathrm{\text{tr}}(P\circ \mathrm{\text{hess}}),\end{array}$$

where the tensor $P:𝔛\left(\Sigma \right)\to 𝔛\left(\Sigma \right)$ is given by

$$P=\sum _{k=r}^s(k+1)c_k^{-1}{b}_k{P}_k.$$

Since $(k+1)c_k^{-1}{b}_{k}0$ for every $k=r,\dots ,s$ and each operator $L_k$ is elliptic (equivalently, each $P_k$ is positive definite) we see that the operator $P$ is positive definite and, consequently, the operator $L$ is elliptic too. So, equation (3.4) and the fact that ${\Sigma }^n$ is $(r,s)$-linear Weingarten imply that

$$L\phi \ge 0.$$

By using once more the weak maximum principle for the elliptic operator $L$ we get

$$\phi \le \underset{\partial \Sigma }{max}\phi \le c{t}_0,$$

that is,

$$c(h-t_0)\le 1.$$

Therefore, we conclude that

$$h\le t_0+\frac{1}{min{H}_1}.$$

This proves $\left(a\right)$.

In the case $minh\ne t_0$, we choose an interior point $q_0$ of ${\Sigma }^n$ satisfying $minh=h\left(q_0\right)$ and the orientation so that $\Theta \ge 0$. Then,

$$0\le \mathrm{\text{Hess}}h\left(q_0\right)(v,v)=\Theta \left(q_0\right)⟨Av,v⟩\left(q_0\right),\forall v\in T_{q_0}\Sigma ,$$

that is, at $q_0$ all the principal curvatures must be nonnegative. So, reasoning as in the previous case we see that each operator $L_k$ is elliptic for every $k=r,\dots ,s$, $H_{k+1}$ is positive on ${\Sigma }^n$ for every $0\le k\le s$ and $h\le t_0$ on ${\Sigma }^n$.

Besides, keeping the notation of case $\left(a\right)$, it follows that $\phi =ch+\Theta$ satisfies, by equations (3.2), (3.3) and our assumption on $K_M$,

$$L_k\phi \le -\frac{c_k}{k+1}⟨\nabla H_{k+1},\nabla h⟩,$$

which implies that $L\phi \le 0$. Therefore, by weak maximum principle we conclude that

$$\phi \ge \underset{\partial \Sigma }{min}\phi \ge c{t}_0,$$

that is,

$$h\ge t_0-\frac{1}{min{H}_1}.$$

This finishes the proof of the theorem. ◻

Remark 1. We observe that the estimate given in Theorem 1 is sharp in the sense that it is reached by the hemisphere ${\Sigma }_{+}=\{x\in {𝕊}^n;x_1\ge 0\}$ of the standard sphere ${𝕊}^n$ in ${ℝ}^{n+1}$. Indeed, it follows easily that ${\Sigma }_{+}$ is a totally umbilical hypersurface (in fact, it is a vertical graph) with $H_1=1$, boundary $\{0\}\times {𝕊}^{n-1}\subset \{0\}\times {ℝ}^n$ and has the maximum height 1.

We observe that Theorem 1 above does not follow from Theorem 1 of de Lima & de Lima 2018, because there the authors just assume that $H_{s+1}0$ and, with this constraint, it is not possible to obtain item (b) above.

Let us also point out that when $s=r$, that is, the hypersurface has constant $(r+1)$-mean curvature $H_{r+1}$, Theorem 1 improves the estimate obtained in Theorem 4.1(i) of Cheng & Rosenberg 2005. Indeed, it is easy to see that the inequality

$$\frac{1}{min{H}_1}\le \frac{1}{H_{r+1}^{1/(r+1)}}$$

holds for every $r=0,\dots ,n-1$. Moreover, this result is also an extension of Theorem 3.5 of Alías & Dajczer 2007 (case $\alpha =0$) and Proposition 1 of Hoffman et al. 2006 (case $\tau =0$).

It is still worth pointing out that the same argument done in the proof of Theorem 4.2 of Cheng & Rosenberg 2005 by using the Alexandrov reflection technique, enable us to get the following consequence of Theorem 1 concerning compact embedded $(r,s)$-linear Weingarten hypersurfaces:

Corollary 1. Let ${\overline{M}}^{n+1}=ℝ\times M^n$ be a product space whose the fiber $M^n$ has nonnegative sectional curvature $K_M$. Let $\psi :{\Sigma }^n\to {\overline{M}}^{n+1}$ be a compact embedded $(r,s)$-linear Weingarten two-sided hypersurface with $(s+1)$-mean curvature $H_{s+1}\ne 0$ on ${\Sigma }^n$, for some $0\le s\le n-1$. Suppose that the angle function $\Theta$ does not change sign on ${\Sigma }^n$. Then ${\Sigma }^n$ is symmetric about some slice $\{t_0\}\times M^n$, $t_0\in ℝ$, and the extrinsic vertical diameter of ${\Sigma }^n$ is no more than $\frac{2}{min\left|H_1\right|}$.

Proceeding, we are able to relax the assumption on the sectional curvature $K_M$ of the fiber $M^n$ letting it be bounded from below by a negative constant. For this, we will assume that the mean curvature satisfies a certain condition, which holds automatically when the sectional curvature of the fiber is nonnegative. In what follows, we will denote by $c=min{H}_1$. So, we get the following result.

Theorem 2. Let ${\overline{M}}^{n+1}=ℝ\times M^n$ be a product space whose the fiber $M^n$ has sectional curvature satisfying $K_M\ge -\alpha$, for some positive constant $\alpha \in ℝ$. Let $\psi :{\Sigma }^n\to {\overline{M}}^{n+1}$ be a compact $(r,s)$-linear Weingarten two-sided hypersurface with $(s+1)$-mean curvature $H_{s+1}\ne 0$ on ${\Sigma }^n$, for some $0\le s\le n-1$, and boundary $\partial {\Sigma }^n$ contained into the slice $\{t_0\}\times M^n$ for some $t_0\in ℝ$. Suppose that the angle function $\Theta$ does not change sign on $\Sigma$ and $c(r+1)min{H}_{k+1}\alpha (s+1)max{H}_k$ for every $k=r,\dots ,s$. Then,

• (a) Either $maxh\ne t_0$ and

  $${\Sigma }^n\subset [t_0,t_0+\frac{(r+1)d}{(r+1)dc-(s+1)\alpha \beta }]\times M^n,$$
  where$d$is given by (3.1) and $\beta ={\sum }_{k=r}^s{b}_{k}max{H}_k$.

• (b) or $minh\ne t_0$ and

  $${\Sigma }^n\subset [t_0-\frac{(r+1)d}{(r+1)dc-(s+1)\alpha \beta },t_0]\times M^n,$$
  where$d$ is given by (3.1) and$\beta ={\sum }_{k=r}^s{b}_{k}max{H}_k$.

Remark 2. We note that in the case of hypersurfaces having constant $(r+1)$-mean curvature $H_{r+1}$, our assumption on $c$ in Theorem 2 becomes $c{H}_{r+1}\alpha max{H}_r$. In particular, it is weaker than assumption (7.77) of Theorem 7.19 of Alías et al. 2016. Moreover, the constant $\frac{(r+1)d}{(r+1)dc-(s+1)\alpha \beta }$ is just given by $\frac{H_{r+1}}{c{H}_{r+1}-\alpha max{H}_r}$. Furthermore, by Lemma 1 we have $c\ge H_{r+1}^{1/(r+1)}$, which implies that

$$\frac{H_{r+1}}{c{H}_{r+1}-\alpha max{H}_r}\le \frac{H_{r+1}}{H_{r+1}^{(r+2)/(r+1)}-\alpha max{H}_r}.$$

In this setting, our estimate improves that one given in Theorem 7.19 of Alías et al. 2016 for the compact case.

On the other hand, we also observe that, since $K_M$ can be negative in Theorem 2, it does not follow from Theorem 1 of de Lima & de Lima 2018.

Proof of Theorem 2. In what follows, we keep the notations established in Theorem 1. Let us suppose $maxh\ne t_0$ first. Then, as in the previous result, taking the angle function $\Theta$ nonpositive, it is easy to see that the operator $L_k$ is elliptic for every $k=r,\dots ,s$, the $(k+1)$-mean curvature $H_{k+1}$ is positive for every $0\le k\le s$ and $h\ge t_0$. Besides, by equations (3.4) and (3.3) we get that the function $\phi$ defined in Theorem 1 satisfies

$$L_k\phi \ge -\frac{c_k}{k+1}⟨\nabla H_{k+1},\nabla h⟩-\Theta \sum _{i=1}^n{\mu }_{i,k}{K}_M(N^{*},E_i^{*}){|N^{*}\wedge E_i^{*}|}^2.$$(3.5)

Since the eigenvalues ${\mu }_{i,k}$ are all positive on ${\Sigma }^n$ and using our assumption on $K_M$ we have

$${\mu }_{i,k}{K}_M(N^{*},E_i^{*}){|N^{*}\wedge E_i^{*}|}^2\ge -{\mu }_{i,k}\alpha {|N^{*}\wedge E_i^{*}|}^2,$$(3.6)

for every $i=1,\dots ,n$ and $k=r,\dots ,s$. With a straightforward computation, we can show that

$${|N^{*}\wedge E_i^{*}|}^2={\left|N^{*}\right|}^2{\left|E_i^{*}\right|}^2-⟨N^{*},E_i^{*}{⟩}^2={\left|\nabla h\right|}^2-⟨E_i,\nabla h{⟩}^2\le 1,$$

which jointly with (3.6) imply that

$$\sum _{i=1}^n{\mu }_{i,k}{K}_M(N^{*},E_i^{*}){|N^{*}\wedge E_i^{*}|}^2\ge -\alpha \mathrm{\text{tr}}{P}_k=-\alpha c_k{H}_k\ge -\alpha c_{k}max{H}_k.$$

From here and (3.5) we infer that

$$L\phi \ge \sum _{k=r}^s(k+1)\alpha \Theta b_{k}max{H}_k\ge (s+1)\alpha \beta \Theta ,$$(3.7)

where $\beta ={\sum }_{k=r}^s{b}_{k}max{H}_k$. On the other hand, by using Proposition 1 we get that

$$Lh=\sum _{k=r}^s(k+1)c_k^{-1}{b}_k{L}_{k}h=\sum _{k=r}^s(k+1)\Theta b_k{H}_{k+1}\le (r+1)d\Theta .$$(3.8)

So, let us consider on ${\Sigma }^n$ the smooth function given by

$$\tilde{\phi }=\phi -\frac{(s+1)\alpha \beta }{(r+1)d}h=\frac{(r+1)dc-(s+1)\alpha \beta }{(r+1)d}h+\Theta .$$

Then, equations (3.7) and (3.8) yield

$$L\tilde{\phi }=L\phi -\frac{(s+1)\alpha \beta }{(r+1)d}Lh\ge (s+1)\alpha \beta \Theta -(s+1)\alpha \beta \Theta =0.$$

Hence, we can apply once more the weak maximum principle to conclude that

$$\tilde{\phi }\le \underset{\partial \Sigma }{max}\tilde{\phi }\le \frac{(r+1)dc-(s+1)\alpha \beta }{(r+1)d}{t}_0,$$

that is,

$$\frac{(r+1)dc-(s+1)\alpha \beta }{(r+1)d}(h-t_0)\le 1.$$(3.9)

Now, the assumption on $c$ gives

$$\begin{array}{ccc}\hfill (r+1)dc-(s+1)\alpha \beta & =\hfill & \hfill (r+1)c\sum _{k=r}^s{b}_k{H}_{k+1}-(s+1)\alpha \sum _{k=r}^s{b}_{k}max{H}_k\\ \hfill & =\hfill & \hfill \sum _{k=r}^s{b}_k((r+1)c{H}_{k+1}-(s+1)\alpha max{H}_k)0.\end{array}$$

Therefore, from equation (3.9) we arrive to

$$h\le t_0+\frac{(r+1)d}{(r+1)dc-(s+1)\alpha \beta }$$

as desired.

Finally, the case $minh\ne t_0$ follows as above and this finishes the proof of the theorem. ◻

As consequence of Theorem 2, the analogue of Corollary 1 also holds in this situation:

Corollary 2. Let ${\overline{M}}^{n+1}=ℝ\times M^n$ be a product space whose the fiber $M^n$ has sectional curvature satisfying $K_M\ge -\alpha$, for some positive constant $\alpha \in ℝ$. Let $\psi :{\Sigma }^n\to {\overline{M}}^{n+1}$ be a compact embedded $(r,s)$-linear Weingarten two-sided hypersurface with $(s+1)$-mean curvature $H_{s+1}\ne 0$ on ${\Sigma }^n$, for some $0\le s\le n-1$. Suppose that the angle function $\Theta$ does not change sign on $\Sigma$ and $c(r+1)min{H}_{k+1}\alpha (s+1)max{H}_k$ for every $k=r,\dots ,s$. Then ${\Sigma }^n$ is symmetric about some slice $\{t_0\}\times M^n$, $t_0\in ℝ$, and the extrinsic vertical diameter of ${\Sigma }^n$ is no more than $\frac{2(r+1)d}{(r+1)dc-(s+1)\alpha \beta }$.

4 - HALF-SPACE THEOREMS

The aim of this section is to give nonexistence results, in the form of half-space theorems, concerning complete two-sided hypersurfaces in the product Riemannian manifold $ℝ\times M^n$. We point out that our results do not assume that some higher order mean curvature of the hypersurface is constant. In this setting, when the fiber $M^n$ is compact, our results generalize those one obtained by Cheng & Rosenberg 2005 and Hoffman et al. 2006 for the case in which the mean curvature or some higher order mean curvature is constant (see Theorems 3 and 4 below). Moreover, in the case in which $M^n$ is not necessarily compact, by using a generalized version of the Omori-Yau maximum principle for trace type differential operators, we prove other interesting half-space theorem (see Theorem 5 below).

According to Hoffman et al. 2006, we say that a two-sided hypersurface in a product space $ℝ\times M^n$ lies in an upper or lower half-space if it is, respectively, contained in a region of $ℝ\times M^n$ of the form

$$[a,+\mathrm{\infty })\times M^n\text{or}(-\mathrm{\infty },a]\times M^n,$$

for some real number $a\in ℝ$.

As an application of Theorem 1 we get the following result:

Theorem 3. Let ${\overline{M}}^{n+1}=ℝ\times M^n$ be a product space whose fiber $M^n$ is compact and has nonnegative sectional curvature. Let $\psi :{\Sigma }^n\to {\overline{M}}^{n+1}$ be a noncompact $(r,s)$-linear Weingarten two-sided properly immersed hypersurface with $(s+1)$-mean curvature bounded away from zero, for some $0\le s\le n-1$, and such that its angle function does not change sign. Then, ${\Sigma }^n$ cannot lie in a half-space.

Proof. Let us assume by contradiction that ${\Sigma }^n$ lies in an upper half-space, that is, ${\Sigma }^n\subset [a,+\infty )\times M^n$, for some $a\in ℝ$. For any number $t_{0}a$, we denote by ${\Sigma }_{t_0}$ the hypersurface

$${\Sigma }_{t_0}=\{(t,p)\in {\Sigma }^n;t\le t_0\}.$$

Then, ${\Sigma }_{t_0}$ is a compact $(r,s)$-linear Weingarten two-sided hypersurface with boundary contained into the slice $\{t_0\}\times M$ and $minh\ne t_0$, because $M^n$ is compact and the immersion is proper. Hence, by Theorem 1 we must have $H_{s+1}0$ on ${\Sigma }_{t_0}$ and ${\Sigma }_{t_0}\subset [t_0-\frac{1}{c\left(t_0\right)},t_0]\times M^n$, where $c\left(t_0\right)={min}_{{\Sigma }_{t_0}}{H}_{1}0$, that is,

$$t_0-a\le \frac{1}{c\left(t_0\right)}.$$

Because $H_{s+1}$ is bounded away from zero we get $inf{H}_{s+1}0$, which implies $inf{H}_{1}0$. Thus

$$t_0-a\le \frac{1}{c\left(t_0\right)}\le \frac{1}{inf{H}_1}.$$

Then choosing $t_0$ large enough we reached a contradiction.

Finally, if ${\Sigma }^n$ is contained into a lower half-space, we may apply the same argument above to arrive at a contradiction. ◻

Similarly, we can reason as in Theorem 3 to obtain as consequence of Theorem 2 the following result, where we keep the notation $c=min{H}_1$.

Theorem 4. Let ${\overline{M}}^{n+1}=ℝ\times M^n$ be a product space whose fiber $M^n$ is compact with sectional curvature satisfying $K_M\ge -\alpha$, for some positive constant $\alpha \in ℝ$. Let $\psi :{\Sigma }^n\to {\overline{M}}^{n+1}$ be a noncompact $(r,s)$-linear Weingarten two-sided properly immersed hypersurface with bounded away from zero $(s+1)$-mean curvature, for some $0\le s\le n-1$, and such that its angle function does not change sign. Suppose that $c(r+1)min{H}_{k+1}\alpha (s+1)max{H}_k$ for every $k=r\dots ,s$. Then, ${\Sigma }^n$ cannot lie in a half-space.

In order to treat the case in which the fiber is not compact, we will make use of a generalized version of the Omori-Yau maximum principle for trace type differential operators proved in Alías et al. 2016. Let ${\Sigma }^n$ be a Riemannian manifold and let $ℒ=\mathrm{\text{tr}}(𝒫\circ \mathrm{\text{hess}})$ be a semi-elliptic operator, where $𝒫:𝔛\left(\Sigma \right)\to 𝔛\left(\Sigma \right)$ is a positive semi-definite symmetric tensor. Following the terminology introduced by Pigola et al. 2005, we say that the Omori-Yau maximum principle holds on ${\Sigma }^n$ for the operator $ℒ$ if, for any function $u\in C^2\left(\Sigma \right)$ with $u^{*}=supu+\infty$, there exists a sequence of points $\left(p_j\right)\subset {\Sigma }^n$ satisfying

$$u\left(p_j\right)u^{*}-\frac{1}{j},\left|\nabla u\left(p_j\right)\right|\frac{1}{j}\mathrm{\text{and}}ℒu\left(p_j\right)\frac{1}{j}$$

for every $j\in \mathbb{N}$. Equivalently, for any smooth function $u\in C^2\left(\Sigma \right)$ with $u_{*}=infu-\infty$ there exists a sequence of points $\left(p_j\right)\subset {\Sigma }^n$ satisfying

$$u\left(p_j\right)u_{*}+\frac{1}{j},\left|\nabla u\left(p_j\right)\right|\frac{1}{j}\mathrm{\text{and}}ℒu\left(p_j\right)-\frac{1}{j}$$

for every $j\in \mathbb{N}$.

We quote a suitable version of the Omori-Yau maximum principle for trace type differential operators on a complete noncompact Riemannian manifold (see Theorem 6.13 of Alías et al. 2016).

Lemma 4. Let ${\Sigma }^n$ be a complete noncompact Riemannian manifold; let $o\in {\Sigma }^n$ be a reference point and denote by $r_o$ the Riemannian distance function from $o$. Assume that the sectional curvature of ${\Sigma }^n$ satisfies

$$K_{\Sigma }\ge -G^2\left(r_o\right),$$(4.1)

with $G\in C^1([0,+\infty ))$ satisfying

$$G\left(0\right)0,G\prime \left(t\right)\ge 0\mathrm{\text{and}}\frac{1}{G\left(t\right)}\notin L^1(+\infty ).$$(4.2)

Let $𝒫$ be a positive semi-definite symmetric tensor on ${\Sigma }^n$. If $sup\mathrm{\text{tr}}\left(𝒫\right)+\infty$, then the Omori-Yau maximum principle holds on ${\Sigma }^n$ for the semi-elliptic operator $ℒ=\mathrm{\text{tr}}(𝒫\circ \mathrm{\text{hess}})$.

In particular, Lemma 4 remains true if we replace condition (4.1) by the stronger condition of ${\Sigma }^n$ having sectional curvature bounded from below by a constant.

Remark 3. As it is well known, especially significant examples of functions $G$ satisfying the condition (4.2) in Lemma 4 are given by (see, for instance Alías et al. 2016 and Pigola et al. 2005)

$$G\left(t\right)=t\prod _{j=1}^N{log}^j\left(t\right),t\gg 1,$$

where ${log}^j$ stands for the $j$-th iterated logarithm.

Now, we are in ready to state and prove our last half-space theorem.

Theorem 5. Let ${\overline{M}}^{n+1}=ℝ\times M^n$ be a product space whose fiber $M^n$ has sectional curvature satisfying $K_M\ge -\alpha$, for some positive constant $\alpha \in ℝ$. Let $\psi :{\Sigma }^n\to {\overline{M}}^{n+1}$ be a complete noncompact $(r,s)$-linear Weingarten two-sided hypersurface with positive $(s+1)$-mean curvature, for some $1\le r\le s\le n-1$. Suppose that $sup\left|H_r\right|+\infty$ and, if $s\ge 2$, there exists an elliptic point in ${\Sigma }^n$. Assume further that the shape operator satisfies $\left|A\right|\le G\left(r_o\right)$, where $G\in C^1([0,+\infty ))$ satisfies (4.2) and $r_o$ is the distance function from a reference point of ${\Sigma }^n$. The following holds:

• (a) either $sup\Theta 0$ or ${\Sigma }^n$ cannot lie in an upper half-space;

• (b) either $inf\Theta 0$ or ${\Sigma }^n$ cannot lie in a lower half-space.

Proof. We begin stating that the sectional curvature $K_{\Sigma }$ of $\Sigma$ satisfies the assumption (4.1) of Lemma 4. Indeed, denoting by $\overline{K}$ the sectional curvature of the ambient space, it follows from Gauss equation (2.1) that if $\{X,Y\}$ is an orthonormal basis for an arbitrary plane tangent to ${\Sigma }^n$, then

$$\begin{array}{ccc}\hfill K_{\Sigma }(X,Y)& =\hfill & \hfill \overline{K}(X,Y)+⟨AX,X⟩⟨AY,Y⟩-⟨AX,Y{⟩}^2\\ \hfill & \ge \hfill & \hfill \overline{K}(X,Y)-\left|AX\right|\left|AY\right|-{\left|AX\right|}^2\\ \hfill & \ge \hfill & \hfill \overline{K}(X,Y)-2{\left|A\right|}^2,\end{array}$$(4.3)

where the last inequality follows from the fact that

$${\left|AX\right|}^2\le \mathrm{\text{tr}}\left(A^2\right){\left|X\right|}^2={\left|A\right|}^2$$

for every unitary vector $X$ tangent to ${\Sigma }^n$. Taking into account that

$$\overline{K}(X,Y)=K_M(X^{*},Y^{*}){|X^{*}\wedge Y^{*}|}^2$$

we obtain from our hypothesis on $K_M$ that $\overline{K}(X,Y)\ge -\alpha$, because ${|X^{*}\wedge Y^{*}|}^2\le {|X\wedge Y|}^2\le 1$. Hence, since the shape operator satisfies $\left|A\right|\le G\left(r_o\right)$, equation (4.3) yields

$$K_{\Sigma }\ge -\alpha -2G^2\left(r_o\right),$$

which concludes the claim.

We prove part (a) first. To do this, we assume that $\Theta \le 0$ and argue by contradiction, that is, we suppose that ${\Sigma }^n$ lies in an upper half-space. Equivalently, the height function of ${\Sigma }^n$ satisfies $h_{*}=infh-\infty$.

We set the second order linear differential operator $ℒ:C^{\infty }\left(\Sigma \right)\to C^{\infty }\left(\Sigma \right)$ by

$$ℒ=\sum _{k=r}^s{c}_k^{-1}{b}_k{L}_k=\mathrm{\text{tr}}(𝒫\circ \mathrm{\text{hess}}),$$

where the tensor $𝒫:𝔛\left(\Sigma \right)\to 𝔛\left(\Sigma \right)$ is given by

$$𝒫=\sum _{k=r}^s{c}_k^{-1}{b}_k{P}_k.$$

Since ${\Sigma }^n$ has an elliptic point (that is, all the principal curvatures are positive in such a point), the operator $L_k$ is elliptic for every $k=r,\dots ,s$ or, equivalently, $P_k$ is positive definite. Then, $𝒫$ is a positive linear combination of the $P_k$’s, so that it is positive definite. Thus, $ℒ$ is a trace type elliptic operator. Besides, by using the identity $\mathrm{\text{tr}}\left(P_k\right)=c_k{H}_k$ we obtain from Lemma 1 that

$$\mathrm{\text{tr}}\left(𝒫\right)=\sum _{k=r}^s{b}_k{H}_k\le \sum _{k=r}^s{b}_k{H}_r^{k/r},$$

which implies that $sup\mathrm{\text{tr}}\left(𝒫\right)+\infty$. Hence, we are ready to apply Lemma 4 to guarantee that the Omori-Yau maximum principle holds on ${\Sigma }^n$ for the operator $ℒ$. Then, there exists a sequence of points $\left(p_j\right)\subset {\Sigma }^n$ having the following properties:

$$limh(p_j)=h_{\ast },|\mathrm{\nabla }h(p_j)|\frac{1}{j}\text{a}nd\mathcal{L}h(p_j)-\frac{1}{j}.$$

In particular, by Proposition 1 we get

$$-\frac{1}{j}ℒh\left(p_j\right)=\sum _{k=r}^s\Theta \left(p_j\right)b_k{H}_{k+1}\left(p_j\right)=d\Theta \left(p_j\right).$$

Since ${\left|\nabla h\right|}^2=1-{\Theta }^2$, we see that $\Theta \left(p_j\right)\to -1$. So, taking limits we conclude that $d\le 0$, which gives a contradiction.

In case (b), we reason again by contradiction, that is, by assuming that $\Theta \ge 0$ and ${\Sigma }^n$ is contained into a lower half-space so that the height function satisfies $h^{*}=suph+\infty$. Then, reasoning as in part (a), it is not difficult to see once more that $d\le 0$, characterizing a contradiction. This concludes the proof of the theorem. ◻

Let us observe that the proof of Theorem 5 remains true with the stronger assumption that $K_{\Sigma }$ is bounded from below by a constant, which implies the validity of the Omori-Yau’s maximum principle. For instance, reasoning as in the proof of Theorem 5 we see that $K_{\Sigma }$ is bounded from below since $sup{\left|A\right|}^2+\infty$. On the other hand, the hypothesis on $H_r$ in Theorem 5, $sup\left|H_r\right|\infty$, can be replaced by $sup{H}_1\infty$, because of Lemma 1. In this case, taking into account the relation

$${\left|A\right|}^2=n^2{H}_1^2-n(n-1)H_2,$$

it follows that the condition $sup{\left|A\right|}^2+\infty$ is equivalent to $sup{H}_1+\infty$. This proves the following result:

Corollary 3. Let ${\overline{M}}^{n+1}=ℝ\times M^n$ be a product space whose fiber $M^n$ has sectional curvature satisfying $K_M\ge -\alpha$, for some positive constant $\alpha \in ℝ$. Let $\psi :{\Sigma }^n\to {\overline{M}}^{n+1}$ be a complete noncompact $(r,s)$-linear Weingarten two-sided hypersurface with positive $(s+1)$-mean curvature, for some $1\le r\le s\le n-1$. Suppose that $sup\left|H_1\right|+\infty$ and, if $s\ge 2$, there exists an elliptic point in ${\Sigma }^n$. The following holds:

• (a) either $sup\Theta 0$ or ${\Sigma }^n$ cannot lie in an upper half-space;

• (b) either $inf\Theta 0$ or ${\Sigma }^n$ cannot lie in a lower half-space.

In the case of hypersurfaces having constant mean curvature the assumption of existence of an elliptic point can be dropped as follows.

Corollary 4. Let ${\overline{M}}^{n+1}=ℝ\times M^n$ be a product space whose fiber $M^n$ has sectional curvature satisfying $K_M\ge -\alpha$, for some positive constant $\alpha \in ℝ$. Let $\psi :{\Sigma }^n\to {\overline{M}}^{n+1}$ be a complete noncompact two-sided hypersurface with positive constant mean curvature and such that $inf{H}_2-\infty$. The following holds:

• (a) either $sup\Theta 0$ or ${\Sigma }^n$ cannot lie in an upper half-space;

• (b) either $inf\Theta 0$ or ${\Sigma }^n$ cannot lie in a lower half-space.

In other words we have:

• (a’) There is no complete noncompact two-sided hypersurface having positive constant mean curvature, angle function nonpositive and contained into an upper half-space;

• (b’) There is no complete noncompact two-sided hypersurface having positive constant mean curvature, angle function nonnegative and contained into a lower half-space.

Proof. It is enough to prove part (a). We suppose that $\Theta \le 0$ and let us reason by contradiction, that is, $infh=h_{*}-\infty$. As in the proof of Theorem 5 and by remark above, we might see that the Omori-Yau maximum principle holds on ${\Sigma }^n$ for the Laplacian. Then, there is a sequence of points $\left(p_j\right)\subset {\Sigma }^n$ satisfying

$$limh(p_j)=h_{\ast },|\mathrm{\nabla }h|\frac{1}{j}\text{and}\mathrm{\Delta }h(p_j)-\frac{1}{j}.$$

By applying Proposition 1 we find

$$-\frac{1}{j}\Delta h\left(p_j\right)=n\Theta \left(p_j\right)H_1$$

Since the angle function is nonpositive, taking limits here we conclude that $H_1\le 0$, which gives a contradiction. ◻

More generally, for hypersurfaces having some constant higher order mean curvature we get the following result:

Corollary 5. Let ${\overline{M}}^{n+1}=ℝ\times M^n$ be a product space whose fiber $M^n$ has sectional curvature satisfying $K_M\ge -\alpha$, for some positive constant $\alpha \in ℝ$. Let $\psi :{\Sigma }^n\to {\overline{M}}^{n+1}$ be a complete noncompact two-sided hypersurface with positive constant $(r+1)$-mean curvature, for some $1\le r\le n-1$. Suppose that $sup{H}_1+\infty$ and, if $r\ge 2$, there exists an elliptic point in ${\Sigma }^n$. The following holds:

• (a) either $sup\Theta 0$ or ${\Sigma }^n$ cannot lie in an upper half-space;

• (b) either $inf\Theta 0$ or ${\Sigma }^n$ cannot lie in a lower half-space.

In other words we have:

• (a’) there is no complete noncompact two-sided hypersurface having $H_{r+1}0$, an elliptic point, with $sup{H}_1+\infty$, angle function nonpositive and contained into an upper half-space;

• (b’) there is no complete noncompact two-sided hypersurface having $H_{r+1}0$, an elliptic point, with $sup{H}_1+\infty$, angle function nonnegative and contained into a lower half-space.

Finally we collect (a) and (b) in the previous corollaries in order to obtain the following result.

Corollary 6. Let ${\overline{M}}^{n+1}=ℝ\times M^n$ be a product space whose fiber $M^n$ has sectional curvature satisfying $K_M\ge -\alpha$, for some positive constant $\alpha \in ℝ$. Let $\psi :{\Sigma }^n\to {\overline{M}}^{n+1}$ be a complete noncompact two-sided hypersurface with positive constant $(r+1)$-mean curvature, for some $0\le r\le n-1$. Suppose that $sup{H}_1+\infty$ and, if $r\ge 2$, there exists an elliptic point in ${\Sigma }^n$. In addition, if $r=0$ assume that $inf{H}_2\ge -\infty$. Then, either $\Theta$ does not vanishes identically or ${\Sigma }^n$ cannot lie in a half-space. In other words, there is no complete noncompact two-sided hypersurface having $H_{r+1}0$, an elliptic point, with $sup{H}_1+\infty$, angle function vanishes identically and contained into a half-space.

ACKNOWLEDGMENTS

The second author is partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil, grant 301970/2019-0. The authors would like to thank the referee for reading the manuscript in great detail and for his/her valuable suggestions and useful comments which improved the paper.

• ALEDO JA, ESPINAR JM & GÁLVEZ JA. 2008. Height estimates for surfaces with positive constant mean curvature in ${𝕄}^2\times ℝ$ Illinois J Math 52(1): 203–211.
• ALEDO JA, ESPINAR JM & GÁLVEZ JA. 2010. The Codazzi equation for surfaces. Adv Math 224(6): 2511-2530.
• ALÍAS LJ & DAJCZER M. 2007. Constant mean curvature hypersurfaces in warped product spaces. Proc Edinb Math Soc 50: 511–526.
• ALÍAS LJ, IMPERA D & RIGOLI M. 2013. Hypersurfaces of constant higher order mean curvature in warped products. Trans Am Math Soc 365(2): 591-621.
• ALÍAS LJ, MASTROLIA P & RIGOLI M. 2016. Maximum Principles and Geometric Applications. Springer: Monographs in Mathematics, 595 p.
• BARBOSA JLM & COLARES AG. 1997. Stability of hypersurfaces with constant r-mean curvature. Ann Global Anal Geom 15: 277–297.
• CHENG X & ROSENBERG H. 2005. Embedded positive constant $r$-mean curvature hypersurfaces in $M^m\times ℝ$ An Acad Bras Cienc 77: 183–199.
• DE LIMA EL & DE LIMA HF. 2018. Height estimates and topology at infinity of hypersurfaces immersed in a certain class of warped products. Aequat Math 92: 737–761.
• ELBERT MF. 2002. Constant positive 2-mean curvature hypersurfaces. Illinois J Math 46: 247–267.
• ESPINAR JM, GÁLVEZ JA & ROSENBERG H. 2009. Complete surfaces with positive extrinsic curvature in product spaces. Comment Math Helv 84(2): 351–386.
• HARDY G, LITTLEWOOD JE & PÓLYA G. 1989. Inequalities. 2nd ed. Cambridge Mathematical Library. 338 p.
• HEINZ E. 1969. On the nonexistence of a surface of constant mean curvature with finite area and prescribed rectifiable boundary. Arch Rational Mech Anal 35: 249–252.
• HOFFMAN D, LIRA J & ROSENBERG H. 2006. Constant mean curvature surfaces in $M^2\times ℝ$ Trans Amer Math Soc 358: 491–507.
• KOREVAAR N, KUSNER R, MEEKS W & SOLOMON B. 1992. Constant Mean Curvature Surfaces in Hyperbolic Space. Amer J Math 114: 1–43.
• KOREVAAR N, KUSNER R & SOLOMON B. 1989. The Structure of Complete Embedded Surfaces with Constant Mean Curvature. J Differ Geom 30: 465–503.
• PIGOLA S, RIGOLI M & SETTI AG. 2005. Maximum principles on Riemannian manifolds and applications. Mem American Math Soc 174(822): 109.
• ROSENBERG H. 1993. Hypersurfaces of constant curvature in space forms. Bull Sci Math 117: 211–239.
• ROSENBERG H, SCHULZE F & SPRUCK J. 2013. The Half-Space Property and Entire Positive Minimal Graphs in $M\times ℝ$ J Diff Geom 95: 321–336.

Publication Dates

• Publication in this collection
  03 Dec 2021
• Date of issue
  2021

History

• Received
  20 May 2019
• Accepted
  8 Feb 2020