Spacelike Hypersurfaces in Weighted Generalized Robertson-Walker Space-Times

Abstract

Applying generalized maximum principle and weak maximum principle, we obtain several uniqueness results for spacelike hypersurfaces immersed in a weighted generalized Robertson-Walker (GRW) space-time under suitable geometric assumptions. Furthermore, we also study the special case when the ambient space is static and provide some results by using Bochner’s formula.

1. Introduction

In recent years, spacelike hypersurfaces in Lorentzian manifolds have been deeply studied not only from their mathematical interest, but also from their importance in general relativity.

Particularly, there are many articles that study spacelike hypersurfaces in weighted warped product space-times. A weighted manifold is a Riemannian manifold with a measure that has a smooth positive density with respect to the Riemannian one. More precisely, the weighted manifold associated with a complete -dimensional Riemannian manifold and a smooth function on is the triple , where stands for the volume element of . In this setting, we will take into account the so-called Bakry-Émery Ricci tensor (see [1]) which as an extension of the standard Ricci tensor , which is defined by Therefore, it is natural to extend some results of the Ricci curvature to analogous results for the Bakry-Émery Ricci tensor. Before giving more details on our work we present a brief outline of some recent results related to our one.

In [2], Wei and Wylie considered the complete -dimensional weighted Riemannian manifold and proved mean curvature and volume comparison results on the assumption that the -Bakry-Émery Ricci tensor is bounded from below and or is bounded. Later, Cavalcante et al. [3] researched the Bernstein-type properties concerning complete two-sided hypersurfaces immersed in a weighted warped product space using the appropriated generalized maximum principles. Moreover, [4] obtained new Calabi-Bernstein’s type results related to complete spacelike hypersurfaces in a weighted GRW space-time. More recently, some rigidity results of complete spacelike hypersurfaces immersed into a weighted static GRW space-time are given in [5].

In this paper we study spacelike hypersurfaces in a weighted generalized Robertson-Walker (GRW) space-times. Moreover, a GRW space-time is a space-time regarding a warped product of a negative definite interval as a base, a Riemannian manifold as a fiber, and a positive smooth function as a warped function. Furthermore, there exists a distinguished family of spacelike hypersurfaces in a GRW space-time, that is, the so-called slices, which are defined as level hypersurfaces of the time coordinate of the space-time. Notice that any slice is totally umbilical and has constant mean curvature.

We have organized this paper as follows. In Section 2, we introduce some basic notions to be used for spacelike hypersurfaces immersed in weighted GRW space-times. In Section 3, we prove some uniqueness results of spacelike hypersurface in a weighted GRW space-time under appropriate conditions on the weighted mean curvature and the weighted function by using the generalized Omori-Yau maximum principle or the weak maximum principle. Finally, in Section 4, applying the weak maximum principle, we obtain some rigidity results for the special case when the ambient space is static.

2. Preliminaries

Let be a connected -dimensional oriented Riemannian manifold and be an open interval in endowed with the metric . We let be a positive smooth function. Denote to be the warped product endowed with the Lorentzian metric where and are the projections onto and , respectively. This space-time is a warped product in the sense of [6], with fiber , base , and warping function . Furthermore, for a fixed point , we say that is a slice of . Following the terminology used in [7], we will refer to as a generalized Robertson-Walker (GRW) space-time. Particularly, if the fiber has constant section curvature, it is called a Robertson-Walker (RW) space-time.

Recall that a smooth immersion of an -dimensional connected manifold is called a spacelike hypersurface if the induced metric via is a Riemannian metric on , which will be also denoted for .

In the following, we will deal with two particular functions naturally attached to spacelike hypersurface , namely, the angle (or support) function and the height function , where is a (unitary) timelike vector field globally defined on and is a unitary timelike normal vector field globally defined on .

Let and stand for gradients with respect to the metrics of and , respectively. By a simple computation, we have Therefore, the gradient of on is Particularly, we have where denotes the norm of a vector field on .

Now, we consider that a GRW space-time is endowed with a weighted function , which will be called a weighted GRW space-time . In this setting, for a spacelike hypersurface immersed into , the -divergence operator on is defined by where is a tangent vector field on .

For a smooth function , we define its drifting Laplacian by and we will also denote such an operator as the -Laplacian of .

According to Gromov [8], the weighted mean curvature or -mean curvature of is given by where is the standard mean curvature of hypersurface with respect to the Gauss map .

It follows from a splitting theorem due to Case (see [9] Theorem ) that if a weighted GRW space-time is endowed with a bounded weighted function such that for all timelike vector fields on , then must be constant along . In the same spirit of this result, in the following we will consider weighted GRW space-times whose weighted function does not depend on the parameter ; that is, . Moreover, for simplicity, we will refer to them as .

In the following, we give some technical lemmas that will be essential for the proofs of our main results in weighted GRW space-times (for further details on the proof, see Lemma in [4]).

Lemma 1. Let be a spacelike hypersurface immersed in a weighted GRW spacetime , with height function . Then,

If we denote as the space of the integrable functions on with respect to the weighted volume element , using the relation of and Proposition in [10], we can obtain the following extension of a result in [11].

Lemma 2. Let be a smooth function on a complete weighted Riemannian manifold with weighted function such that does not change sign on . If , then vanishes identically on .

In the following, we will introduce the weak maximum principle for the drifted Laplacian. By the fact in [12], that is, the Riemannian manifold satisfies the weak maximum principle if and only if is stochastically complete, we can have the next lemma which extended a result of [13].

Lemma 3. Let be an -dimensional stochastically complete weighted Riemannian manifold and be a smooth function which is bounded from below on . Then there is a sequence of points such that Equivalently, for any smooth function which is bounded from above on , there is a sequence of points such that

3. Uniqueness Results in Weighted GRW Space-Times

In this section, we will state and prove our main results in weighted GRW space-times . We point out that, to prove the following results, we do not require that the -mean curvature of the spacelike hypersurface is constant.

Recall that a slab of a weighted GRW spacetime is a region of the type

Theorem 4. Let be a weighted GRW spacetime which obeys . Let be a complete spacelike hypersurface that lies in a slab of . If the -mean curvature satisfies and , then is a slice of .

Proof. From (10), we have By the hypotheses, we have . Moreover, since lies in a slab, there is a positive constant such that Therefore, we can apply Lemma 2 to get ; that is, is constant. Therefore is a slice.

Theorem 5. Let be a weighted GRW spacetime which obeys . Let be a complete spacelike hypersurface that lies in a slab of . If the -mean curvature satisfies and , then is a slice of .

Proof. By a similar reasoning as in the proof of Theorem 4, we have where the last inequality is due to .
Taking into account the assumptions, we have . Now in the same argument as in Theorem 4, we have that is a slice.

Next, we will use the weak maximum principle to study the rigidity of the spacelike hypersurfaces in weighted GRW space-times.

Theorem 6. Let be a weighted GRW spacetime which satisfies and there is a point such that . Let be a stochastically complete constant -mean curvature spacelike hypersurface such that , which is contained in a slab; then is -maximal. In addition, if is complete and , then is a slice.

Proof. We take the Gauss map of the hypersurface such that ; from (7) we have .
By Lemma 3, the weak maximum principle for the drifted Laplacian holds on ; then there exist two sequences such that On the other hand, from (9), we have Since lies in a slab, if is bounded from below, then Moreover, if is bounded from above, we get Considering that the function is increasing, then Hence, ; that is, is a -maximal spacelike hypersurface. Using (10), we have In the following, by the same argument as in Theorem 4, we have that is a slice.

4. Weighted Static GRW Space-Times

In this section, we obtain some rigidity results of stochastically complete hypersurfaces in weighted static GRW space-times by the weak maximal principle. Firstly, we give the following technical result which extended the corresponding conclusion in [12].

Lemma 7. Let be a stochastically complete Riemannian manifold and be a nonnegative smooth function on . If there exists a positive constant such that , then .

Theorem 8. Let be a stochastically complete hypersurface with constant -mean curvature in a weighted static GRW spacetime . Assume that for some positive constant and the weighted function is convex. If for some constant , then is a slice.

Proof. Let be a (local) orthonormal frame in ; using the Gauss equation, we have that for . Moreover, we also have where is the sectional curvature of the fiber and and are the projections of the tangent vector fields and onto .
By a direct computation and considering the hypothesis , we get Substituting (25) into (23), Furthermore, taking into account that the weighted function is convex, we have Therefore, In particular, we have Now we recall the Bochner-Lichnerowicz formula (see [2]): From the fact that is a constant, we have By [14], we get Using (29), (31), and (32) in (30), we have Finally, considering the hypothesis , we obtain Thus, there is a positive constant such that Therefore, is constant by Lemma 7.

Theorem 9. Let be a stochastically complete hypersurface with constant -mean curvature in a weighted static GRW space-time . Assume that the sectional curvature is nonnegative and the weighted function is convex. If is bounded from above, then is -maximal.

Proof. As in the proof of Theorem 8, taking into account that the hypothesis is nonnegative, there is a constant such that Moreover, considering the relation , we have Using (9) and (37), we obtain By the hypothesis that is bounded from above, applying Lemma 3, the weak maximum principle, we get Therefore is -maximal.

As a consequence of the proof of Theorem 8, we can get the following corollary.

Corollary 10. Let be a stochastically complete hypersurface with constant -mean curvature in a weighted static GRW space-time . Assume that and for some positive constants and . If for some constant , then is a slice.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by National Natural Science Foundation of China (no. 11371076).