#### Abstract

By applying Omori-Yau maximal principal theory and supposing an appropriate restriction on the norm of gradient of height function, we obtain some new Bernstein-type theorems for complete spacelike hypersurfaces with nonpositive constant mean curvature immersed in a semi-Riemannian warped product. Furthermore, some applications of our main theorems for entire vertical graphs in Robertson-Walker spacetime and for hypersurfaces in hyperbolic space are given.

#### 1. Introduction

The theory of spacelike hypersurfaces immersed in semi-Riemannian warped products with constant mean curvature has got increasing interest both from geometers and physicists recently. One of the basic questions on this topic is the problem of uniqueness for this type of hypersurfaces. The aim of this paper is to study such type problems. Before giving details of our main results, we firstly present a brief outline of some recent papers containing theorems related to ours.

By using a suitable application of well-known generalized maximal principal of Omori [1] and Yau [2], Albujer et al. [3] obtained uniqueness results concerning complete spacelike hypersurfaces with constant mean curvature immersed in a Robertson-Walker spacetime.

By applying a maximal principal due to Akutagawa [4], Aquino and de Lima [5] obtained a new Bernstein-type theorem concerning constant mean curvature complete vertical graphs immersed in a Riemannian warped product, which is supposed to satisfy an appropriate convergence condition defined as the following: where denotes the sectional curvature of the fiber , which is very different from null convergence condition (defined by (2)).

Replacing the null convergence condition by , Alías et al. obtained uniqueness theorems (see Section 2 of [6]) concerning compact spacelike hypersurface with higher constant mean curvature immersed in a spatially closed generalized Robertson-Walker spacetime. We also refer the reader to [7] for the other relevant results.

In this paper, following [6, 8], we consider the Laplacian of integral of warping function. By using a suitable maximal principal of Omori and Yau and supposing an appropriate restriction on the norm of gradient of height function, we obtain some new Bernstein-type theorems as the following.

Theorem 1. *Let be a Robertson-Walker spacetime whose Riemannian fiber has constant sectional curvature satisfying the null convergence condition defined as the following:
**
Let be a complete spacelike hypersurface with constant mean curvature . Suppose that is bounded away from the infinity of and that
**
If the height function of satisfies
**
for some positive constant and , then is a slice.*

Theorem 2. *Let be a Riemannian warped product whose fiber has constant sectional curvature satisfying
**
Let be a complete spacelike hypersurface with constant mean curvature and is bounded from below. Suppose that is bounded away from the infinity of and that
**
If the height function of satisfies
**
for some positive constant and , then is a slice.*

#### 2. Preliminaries

In this section, we recall some basic notations and facts following from [3, 8] that will appear along this paper.

Let be a connected, -dimensional oriented Riemannian manifold, an interval, and a positive smooth function. We consider the product differential manifold and denote by and the projections onto the base and fiber , respectively. A particular class of semi-Riemannian manifolds is the one obtained by furnishing with the metric for any and any , where . We call such a space warped product manifold; is known as the warping function, and we denote the space by . Note that is called a generalized Robertson-Walker spacetime, in particular, is called a Robertson-Walker spacetime if has constant sectional curvature. From [9], we know that a generalized Robertson-Walker spacetime has constant sectional curvature if and only if the Riemannian fiber has constant sectional curvature and the warping function satisfies the following differential equation:

It follows from [10] that the vector field is conformal and closed (in this sense that its dual -form is closed) with conformal factor , where the prime denotes differentiation with respect to . For , we orient the slice by using the unit normal vector field , then from [11] we know has constant th mean curvature with respect to .

A smooth immersion of an -dimensional connected manifold is said to be a spacelike hypersurface if the induced metric via is a Riemannian metric on . If is oriented by the unit vector field , one obviously has . Moreover, when , we may define the normal hyperbolic angle of as being the smooth function given by

We denote by and the Levi-Civita connections in and , respectively. Then the Gauss-Weingarten formulas for the spacelike hypersurface are given by for any , where be the shape operator of with respect to its Gauss map , and denotes the Lie algebra of all tangential vector fields on .

The curvature tensor of a spacelike hypersurface is given by [12] as the following: where denotes the Lie bracket and .

Let and be the curvature tensors of and , respectively. Denote by the tangential component of a vector field ; thus, for any we have the following Gauss equation: Consider a local orthonormal frame and , then the Ricci curvature tensor of is given as the following:

#### 3. Key Lemmas

We consider two particular functions naturally attached to complete spacelike hypersurfaces, namely, the vertical (height) function and the support function . Denote by and the gradients with respect to the metrics of and , respectively. Thus, by a simple computation, we have the gradient of on as the following: Thus, the gradient of on is given by We denote by the norm of a vector field on , then we get

According to [4], a spacelike hypersurface is said to be bounded away from the future infinity of if there exists such that Analogously, a spacelike hypersurface is said to be bounded away from the past infinity of if there exists such that Finally, is said to be bounded away from the infinity of if it is both bounded away from the past and future infinity of .

In order to prove our main theorems, we will make use of the following computations. We also refer the reader to [6, 11] for a more generalized proposition which makes the following lemma as its trivial case.

Lemma 3. *Let be a spacelike hypersurface immersed in a semi-Riemannian warped product. If
**
then
**
where denotes the Laplacian operator and is the height function of . *

* Proof. *Let be the gradient with respect to the metric of induced from . Thus, it is easy to see
Denote by a local orthonormal frame of . By the definition of Laplacian operator and using (22), then we have

By taking in Lemma of [13] (we also refer the reader to Lemma of [11]), then we have
By substituting (24) into (23), we complete the proof.

We need another lemma proved by Albujer et al. in [3].

Lemma 4. *Let be a spacelike hypersurface immersed in a Robertson-Walker spacetime whose Riemannian fiber has constant sectional curvature . Denote by the height function on . If obeys the null convergence condition, then
**
where is the Riemannian curvature tensor of and . *

*Remark 5. *It follows from (14) and Lemma 4 that, if the Riemannian fiber has constant sectional curvature, the Ricci curvature tensor of is bounded from below if is bounded away from the infinity of .

We also need another lemma shown by Aquino and de Lima in [5].

Lemma 6. *Let be a warped product which satisfies convergence condition (5). Let be a complete hypersurface with both mean curvature and second fundamental form bounded. If is bounded on , then the Ricci curvature of is bounded from below. *

In order to prove our main theorems, we also need the well-known generalized maximal principal due to Omori [1] and Yau [2].

Lemma 7. *Let denote an -dimensional complete Riemannian manifold whose Ricci curvature tensor is bounded from below. Then, for any -function with , there exists a sequence of points in satisfying the following properties:
**
Equivalently, for any -function with , there exists a sequence of points in satisfying the following properties:
*

#### 4. Proofs of Main Theorems

* Proof of Theorem 1. *Since is a unitary timelike vector field globally defined on the ambient spacetime, then there exists a unique timelike unitary normal vector filed globally defined on the spacelike hypersurface which is the same time orientation as . By using the reverse Cauchy-Schwarz inequality, we have
We observe that, from Lemma 3,

By using the assumption (3), we have that . Then, it follows from (28) that . Notice that the warping function is positive on , then from (3) and (29), we have the following inequality:

On the other hand, since the spacelike hypersurface is bounded away from the infinity of , and the constant sectional curvature of the fiber satisfies the null convergence condition, then it follows from Lemma 4 and Remark 5 that the Ricci curvature tensor of is bounded from below. Then by the definition of , that is, (20), it is easy to see . Thus, applying Lemma 7 to the smooth function on implies that there exists a sequence of points in with the following properties:
Note that is a nonpositive constant, then it follows from (30) that

Since the warping function is positive on and is bounded away from the infinity, from (31) and (32), we have , which means that

Substituting the above equation into (4) implies that , that is, is a constant on ; thus is a slice of .

From (9), we see that every Robertson-Walker spacetime with constant sectional curvature trivially obeys the null convergence condition (2). Therefore, from Theorem 1, we obtain the following result.

Corollary 8. *Let be a Robertson-Walker spacetime with constant sectional curvature. Let be a complete spacelike hypersurface with constant mean curvature . Suppose that is bounded away from the infinity. If (3) and (4) hold, then is a slice.*

Theorem 9. *Let be a Robertson-Walker spacetime whose Riemannian fiber has constant sectional curvature satisfying the null convergence condition. Let be a complete spacelike hypersurface with constant mean curvature and bounded away from the infinity. If
**
and the height function of satisfies
**
for some positive constant and , then is a slice.*

*Proof. *From (34), we see that the constant mean curvature is nonnegative, then it follows from (28) and (34) that . Thus, from (29), we have

Using the similar analysis with the proof of Theorem 1, applying the Lemma 7 to the smooth function on implies that there exists a sequences of points in satisfying the following properties:
It follows from (36) that
From (37) and (38), we have . Thus, from (36), we get

By substituting the above equation into (35), we have , that is, is a constant on ; thus is a slice.

Corollary 10. *Let be a Robertson-Walker spacetime with constant sectional curvature. Let be a complete spacelike hypersurface with constant mean curvature . Suppose that is bounded away from the infinity. If (34) and (35) hold, then is a slice.*

Noting that a Robertson-Walker spacetime with constant sectional curvature trivially obeys the null convergence condition; thus the proof of Corollary 10 follows from Theorem 9.

It is well known that Robertson-Walker spacetime is called static Robertson-Walker spacetime if the warping function is constant. Without losing the generality, we consider in the following. In this case, the null convergence condition (2) implies that . Also, (3) becomes that . When is a constant, (33) (or (39)) implies that , that is, is maximal. On the other hand, letting the warping function be a constant, we see from (34) that . Then a weaker assumption than (4) and (35), that is, the normal hyperbolic angle is bounded on , also guarantees Theorems 1 and 9. Thus, from Theorems 1 and 9, we have the following corollary.

Corollary 11. *Let be a static Robertson-Walker spacetime whose Riemannian fiber has non-negative constant sectional curvature . Let be a complete, connected spacelike hypersurface with constant mean curvature . Suppose that the normal hyperbolic angle of is bounded, then is a maximal slice.*

We also refer the reader to [3, 14] for the similar results with Corollary 11 proved by using the different methods with ours. Next, we give the proof of Theorem 2, which is the Riemannian warped product version of our Theorem 1.

*Proof of Theorem 2. *Initially, we consider that the orientation of the hypersurface such that its angle function satisfies
Then from Lemma 3, we have that

By using the assumption (6) we see that the mean curvature is non-positive, then it follows from (40) that . Notice that the warping function is positive on , then from (41) we have the following inequality:

Since the spacelike hypersurface is bounded away from the infinity, it follows that is bounded on . Let denote the second elementary symmetric function on the eigenvalues of the shape operator , and denotes the mean value of . It is easy to see . Then the assumption that is bounded from below means that is bounded from above; from Lemma 6 we know that the Ricci curvature tensor of is bounded from below.

Finally, by applying analogous arguments employed in the last part of the proof of Theorem 1, we conclude that is a slice of .

#### 5. Application of Main Theorems

In this section, we consider a particular model of Lorentzian warped product, namely, the steady state space, that is, the warped product

The importance of considering comes from the fact that, in cosmology, is the steady model of the universe proposed by Bondi and Gold [15] and Hoyle [16]. Moreover, in physical context the steady state space appears naturally as an exact solution for the Einstein equations, being a cosmological model where matter is supposed to travel along geodesic normal to horizontal hyperplanes. We also notice that Montiel [17] gave an alternative description of the steady state space as follows.

Let denote the -dimensional Lorentzian-Minkowski space, that is, the real vector space with a Lorentzian metric for all . The -dimensional de Sitter space is defined as the following: . Let be a nonzero null vector of the null cone with vertex in the origin, such that , where . Then, it can be shown that the open region of the de Sitter space is isometric to (see [18] for details).

Recently, some uniqueness theorems for steady state space were obtained by [13, 19]. In fact, Caminha and de Lima [19] proved the following results.

Theorem 12. *Let be a Riemannian immersion of a complete surface of non-negative Guassian curvature with constant mean curvature . If
**
then is a slice of . *

Suppose that the warping function ; thus (34) becomes ; meanwhile the inequality (35) becomes where both and are positive constants. Suppose that , then we have for all if . Thus, our Theorem 9 extends Theorem in [19]. At last, we write the uniqueness theorems for surface in steady state space which follows from Theorems 1 and 9, respectively, as following.

Theorem 13. *Let be a Riemannian immersion of a complete surface with constant mean curvature . If
**
holds for some positive constant and , then is a slice of . *

Theorem 14. *Let be a Riemannian immersion of a complete surface with constant mean curvature . If
**
holds for some positive constant and , then is a slice of . *

#### 6. Entire Vertical Graphs in Robertson-Walker Spacetime

In the last section of this paper, we investigate the applications of our main theorems for entire vertical graphs in a Robertson-Walker spacetime. We follow [3, 8, 20] for the basic notations and facts used in this section.

Let be a connected domain of ; a vertical graph over is defined by smooth function and it is given by The metric induced on from the Lorentzian metric on the ambient space via is . The graph is said to be entire if . It is easy to see that a graph is a spacelike hypersurface if and only if , where is the gradient of in and its norm, both with respect to the metric in .

Let be a spacelike vertical graph over a domain , its future-pointing Gauss map is given by the vector field Moreover, the shape operator of with respect to is given by for any tangent vector field on , where denotes the Levi-Civita connection in with respect to the metric . The mean curvature function of a spacelike graph with respect to is given as the following: where denotes the divergence operator on with respect to the metric . Now, we give the following nonparametric version of Theorem 1.

Theorem 15. *Let be a Robertson-Walker spacetime whose Riemannian fiber has constant sectional curvature . Suppose that obeys the null convergence condition (2). Let be an entire spacelike vertical graph in with constant mean curvature and bounded away from the infinity. If
**
for some positive constant and , then is a slice. *

*Proof. *First of all and following the ideas of Albujer et al. in the proof of Theorem 4.1 in [20] and Theorem 4.1 in [3], it can be easily seen that the induced metric on the entire graph is complete. Then it follows from [8, 20] that
Thus, together with (55) and (56) and by a straightforward computation, we know that (4) holds. Then the proof of Theorem 15 follows from Theorem 1.

Let be a static Robertson-Walker spacetime, that is, . Thus, the null convergence condition implies that and (54) becomes . In particular, from (55), we get

Following from Theorem 15, we get a uniqueness theorem for entire spacelike vertical graphs in static spacetime.

Theorem 16. *Let be a static Robertson-Walker spacetime whose Riemannian fiber has non-negative constant sectional curvature . Let be an entire spacelike vertical graph in with non-positive constant mean curvature and bounded away from the infinity. If the condition (57) is satisfied for some positive constant and , then is a slice. *

#### Acknowledgments

The project is supported by Natural Science Foundation of China (no. 10931005) and Natural Science Foundation of Guangdong Province of China (no. S2011010000471).