Abstract

We study constant mean curvature hypersurfaces constructed over the fiber of warped products . In this setting, assuming that the sign of the angle function does not changed along the hypersurfaces, we infer the uniqueness of such hypersurfaces by applying a parabolicity criterion. As an application, we get some Bernstein type theorems.

1. Introduction

In this paper, we investigate the uniqueness results in a certain class of Riemannian manifolds, that is, the warped products. In the sense of [1], the warped products , where the base is an interval, the -dimensional Riemannian manifold is a fiber, and is a warping function (for further details, see Section 2). Before presenting details on our work, we give a brief overview of some articles concerning ours.

In [2], Montiel proved that any compact orientable constant mean curvature hypersurfaces in warped products that can be written as a graph over must be a slice, under the assumption that the Ricci curvature of and the function satisfy the convergence condition . Later on, Alas and Dajczer [3] studied the constant mean curvature hypersurfaces in warped product spaces. In this setting, if the hypersurface is compact, they extended the previous results by Montiel. Afterwards, some of these generalizations hold for complete hypersurfaces. In recent years, by using the Omori-Yau generalized maximum principle for complete hypersurfaces and supposing suitable assumptions, some researchers proved that such a hypersurface must be a slice. For instance, in [4], Caminha and de Lima studied complete graphs of constant mean curvature in the hyperbolic and steady-state spaces, and they obtained some rigidity theorems for such graphs. Later, Aquino and de Lima [5] extended the results in [4] to complete constant mean curvature graphs in warped products under appropriate convergence condition. In [6], Cavalcante et al. considered the Bernstein type properties of complete two-sided hypersurfaces in weighted warped products; they established sufficient conditions which guarantee that such a hypersurface must be a slice. Furthermore, [7] obtained uniqueness results for complete hypersurfaces in Riemannian warped products whose fiber has parabolic universal covering. More recently, by the weak Omori-Yau’s maximum principle, the author [8] proved new Bernstein type results of complete constant weighted mean curvature hypersurfaces in weighted warped products .

This paper is organized as follows: in Section 2, we introduce some basic facts for hypersurfaces in warped products. Section 3 is devoted to compute the Laplacian of the angle function which we will define in Section 2. Moreover, using the parabolicity criterion, we establish the uniqueness results concerning constant mean curvature hypersurfaces. As a consequence of this previous study, we prove some Bernstein type results for constant mean curvature entire graphs in warped products.

2. Preliminaries

Throughout this paper, we consider the warped products , where is a connected oriented -dimensional Riemannian manifold, is an interval with a positive definite metric , is a positive smooth function, and the product manifold is endowed with the Riemannian metric where and denote the projections onto and , respectively. Such the resulting space is said to be a warped product in [1], Chapter 7, with fiber, base, and warping function. Moreover, an immersion of an -dimensional manifold is called a hypersurface. Furthermore, the induced metric through on is also denoted by .

In fact that each leaf of the foliation of by complete hypersurfaces has constant mean curvature . Here, we say that is a slice of . Thus, a slice is minimal if and only if .

Observe that the vector field is a closed conformal vector field in , that is, where is a unit vector field tangent to , , and is the Levi-Civita connection in .

In this paper, we consider the hypersurfaces oriented a unit normal vector field , and such hypersurfaces are called two-sided hypersurfaces. In what follows, we will study its angle (or support) function and height function.

Let be the Levi-Civita connection in . A direct computation shows that

Thus, the gradient of is given by where is the tangential component of a vector field in along . Moreover, where is the norm of a vector field on .

3. Parametric Uniqueness Results

In order to establish our uniqueness results in warped products , we need to compute the Laplacian of the angle function to obtain a bounded on the Laplacian of the function .

Lemma 1. Let be a hypersurface in a warped product . Then, the angle function of satisfies where is the Ricci curvature tensor of and stands for the projection of the vector field onto .

Proof. The Gauss and Weingarten formulas of are where and is the shape operator of corresponding to .
Now, taking the tangential component in (2), by (7) and (8), we can easily get that where and is the tangential component of along . Therefore, it follows from (9) that Moreover, by (4) and (9), we conclude that Let be a local orthonormal frame on ; from (10), we know that In fact, for every , we have that where denotes the covariant derivative of . From (12) and (13), we can rewrite (12) as Recall that the Codazzi equation of is or, equivalently, where denotes the curvature tensor of . Therefore, using (11) and (15), we conclude from (14) that Here, we know the general fact that for every . Then, it follows from (9) that where stands for the Ricci curvature tensor of . On the other hand, using Corollary 7.43 of [1], we have that where is the Ricci curvature tensor of the fiber and and denote the projections of the vector field and onto , respectively. Moreover, from (4), we obtain Therefore, from (20) and (21), we get Finally, substituting (22) into (19), by a direct computation, we can obtain (6).

We recall that a complete Riemannian manifold is parabolic in the sense that any positive superharmonic function on the Riemannian manifold must be constant (see [9]). In this setting, we obtain some uniqueness results via parabolicity criterion.

Theorem 2. Let be a complete parabolic hypersurface with constant mean curvature in a warped product whose fiber has nonnegative Ricci curvature. Assume that the warping function satisfies . If and , then is totally geodesic with constant warping function. In addition, if the fiber has positive Ricci curvature, then is a totally geodesic minimal slice.

Proof. From (10), we obtain that Therefore, using (6) and (23), we deduce that Moreover, using Young’s inequality, we have So, we can estimate Using the hypothesis of Theorem 2, we conclude that and , which suffices to show that is constant, So, . From (26), we know that It follows that the hypersurface is totally geodesic and is constant. In addition, if the fiber has positive Ricci curvature, from (26), we have that at any ; that is, on , and then, is a totally geodesic slice. Moreover, we note that the mean curvature of a slice in a warped product is given by Therefore, is a totally geodesic minimal slice.

Remark 3. For the two-dimensional case, we can set . More precisely, for any dimension, we can chose , where is the only real (positive) root of ; this root goes to zero as .

On the other hand, if the warped products satisfy the following convergence condition: we have the following.

Theorem 4. Let be a complete parabolic hypersurface with constant mean curvature in a warped product which satisfies the convergence condition (29). Assume that the warping function satisfies . If and , then is totally geodesic with constant warping function. In addition, if the inequality (29) is strict, then is a totally geodesic minimal slice.

Proof. By an analogous way in the proof of Theorem 2, we can estimate Considering condition (29), it follows that . Moreover, since , we get . Under the assumptions of Theorem 4, from (30), we conclude that , taking into account that the hypersurface is parabolic. Via parabolicity criterion, we get that is constant, , and . Therefore, we have that the hypersurface is totally geodesic and is constant.
Moreover, if (29) is strict, by a similar reasoning as in the proof of Theorem 2, we get that is a totally geodesic minimal slice.

Remark 5. Note that by weakening the assumptions of Theorem 2 and Theorem 4, the uniqueness result of warped product does not hold. In fact, the sphere immersed in a slice of with satisfies all hypothesis of both theorems except that , since the mean curvature of the sphere is negative. It is a slice, but it fails to be totally geodesic.
Moreover, if the fiber has nonnegative Ricci curvature, assume that the warping function satisfies which automatically implies that the hypersurface obeys the convergence condition (29). Thus, Theorem 4 extends Theorem 2.

It should also be noticed that the similar idea has been used to obtain the rigidity of hypersurfaces in warped products (see [10], Theorem 4.11). Nevertheless, we take a different approach to prove our main uniqueness results (Theorems 2 and 4) of constant mean curvature hypersurfaces in warped products.

4. Bernstein Type Problems

In the nonparametric case of hypersurfaces in the Riemannian manifold, there is a very celebrated result known as the Bernstein theorem. The original Bernstein theorem is that each complete minimal surface in that can be written as the graph of a function on must be a plane. Later, Chern [11] built a different proof of the original Bernstein theorem. In [12], Simons extended the Bernstein theorem to Euclidean space () in which any complete minimal hypersurface in must be a hyperplane with , the result which is obtained by successive efforts of Almgren [13], Fleming [14], and De Giorgi [15]. However, for , Bombieri et al. [16] constructed a counterexample. In recent years, many researchers made great efforts to extend these Bernstein type theorems to a wide variety of ambient spaces.

As a consequence of the parametric case, in this section, we will solve the Bernstein problem in warped products. Therefore, we study the graph over in the warped product , given by where denote a connected domain of and is a smooth function on . Moreover, the graph inherits from a metric, which is defined by

Note that if is complete and , then is complete. Moreover, for any point , we have . Therefore, and may be naturally identified on .

Moreover, if , then the graph is entire graph. In this case, the unit normal vector field on is

Thus, the mean curvature function of associated to is

In the following, we will apply the uniqueness results of constant mean curvature hypersurfaces obtained in Section 3 to prove new Bernstein type results for the constant mean curvature hypersurface equation:

Theorem 6. Let be a parabolic entire graph in , where is a smooth function which satisfies . Then, the only bounded entire solutions to equation (35) with , for and such that on a complete Riemannian manifold with positive Ricci curvature, are the constant functions , with such that .

Proof. A straightforward computation yields It follows that from the constraints and (36), we have that Finally, applying Theorem 2, the proof ends.

Furthermore, we can use condition in Theorem 4 to prove a Bernstein type result in an analogous way.

Theorem 7. Let be a parabolic entire graph in , where is a smooth function which satisfies . Then, the only bounded entire solutions to equation (35) with , for and such that on a complete Riemannian manifold satisfies the convergence condition (29), are the constant functions , with such that .