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Advances in Mathematical Physics
Volume 2013 (2013), Article ID 959143, 5 pages
On Bernstein-Type Theorems in Semi-Riemannian Warped Products
1Department of Mathematics, South China University of Technology, Guangzhou, Guangdong 510641, China
2School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, China
Received 10 April 2013; Revised 29 July 2013; Accepted 30 July 2013
Academic Editor: Shao-Ming Fei
Copyright © 2013 Wenjie Wang and Ximin Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Complete spacelike hypersurfaces immersed in semi-Riemannian warped products are investigated. By using a technique according to Yau (1976) and a reasonable restriction on the mean curvature of the hypersurfaces, we obtain some new Bernstein-type theorems which extend some known results proved by Camargo et al. (2011) and Colares and Lima (2012).
In the present paper, we are interested in the study of the complete spacelike hypersurfaces immersed in semi-Riemannian warped product manifolds, in particular, in steady state-type spacetime and hyperbolic-type space. Before giving details of our main results, we first present a brief outline of some recent papers containing theorems related to ours.
By using a restriction on the height function of a complete spacelike hypersurface, Caminha and de Lima  obtained some unique results concerning complete spacelike hypersurfaces with constant mean curvature immersed in steady state space and hyperbolic space, respectively.
Later, Albujer and Alías  proved that on a complete spacelike hypersurface the constant mean curvature must be identically 1, provided that such a hypersurface is bounded away from the infinity of the steady state space. For some other Bernstein-type results concerning constant mean curvature, we refer the reader to some recent papers by Albujer et al. [3, 4] and Aquino and de Lima .
Also, by using the well known result according to Yau , Camargo et al. in  obtained some Bernstein-type results concerning complete spacelike hypersurfaces, in steady state-type and hyperbolic-type space. Noticing that in their paper the mean curvature of the complete spacelike hypersurface need not be a constant.
de Lima in  obtained a new Bernstein-type theorem concerning complete spacelike hypersurfaces in hyperbolic space with the bounded mean curvature (not necessarily constant) and a restriction on the normal angle.
In this paper, following [9, 10] we shall consider the Laplacian of the integral of the warping function. In fact, by using a technique provided by Yau in  and supposing an appropriate restriction on the mean curvature, we obtain the following Bernstein-type theorems.
Theorem 1. Let be a generalized Robertson-Walker spacetime whose fiber is a complete Riemannian manifold. Let be a complete and connected spacelike hypersurface with the mean curvature satisfying If has integrable norm on , then is a slice of .
The Riemannian version of Theorem 1 is also presented as follows.
Theorem 2. Let be a Riemannian warped product whose fiber is a complete Riemannian manifold. Let be a complete and connected spacelike hypersurface with the mean curvature satisfying If has integrable norm on , then is a slice of .
Suppose that the warping function in both Theorems 1 and 2 is given by for , then the above two theorems are just the corresponding results shown in . Finally, letting the warping function in Theorem 2 be for and the fiber an -dimensional Euclidean space, then Theorem 2 gives a Bernstein-type theorem for spacelike hypersurfaces immersed in the hyperbolic space (see Section 4 for details).
This paper is organized as follows. We shall first recall some notations and collect some basic facts in a preliminaries section, and some key lemmas used to prove our main theorems are also given in this section. Section 3 is devoted to proving some unique theorems concerning spacelike hypersurfaces in semi-Riemannian warped products. Finally, in Section 4, some applications of our main theorems in steady state-type spacetimes and hyperbolic-type spaces are obtained respectively.
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 product manifolds with the metric for any and any , where . We call such a space warped product manifold, and 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 the fiber has constant sectional curvature. From  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 [14, 15] 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 the unit normal vector field , then from  we know that has constant th mean curvature with respect to for .
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 .
We consider two particular functions naturally attached to complete spacelike hypersurfaces, namely, the vertical (height) function and the support function . We denote by and the gradients with respect to the metrics of and , respectively. Thus, by a simple computation we present the gradient of on as follows: Moreover, the gradient of on is given by We denote by the norm of a vector field on , then we get
According to [2, 16], 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 .
Setting in Lemma 4.1 of , we may obtain the Laplacian of the integral of the warping function in a generalized Robertson-Walker spacetime. By using the technique according to Alías and Colares , the second author and Wang in  generalize this result in a semi-Riemannian warped product as follows.
Furthermore, we also need the following well known lemma according to Yau .
Lemma 4 (corollary on page 660 of ). Let be an -dimensional complete Riemannian manifold. If is a smooth subharmonic or superharmonic function whose gradient norm is integrable on , then must be actually harmonic.
3. 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
Letting , then it follows from Lemma 3 that
The assumptions of Theorem 1 assure that , then, it follows from (12) that
Noticing that the warping function is positive on , then, by using the above inequality and (1) in (13) we have the following inequality:
this means that is a subharmonic function on .
On the other hand, since the spacelike hypersurface is bounded away from the infinity of , then the height function is bounded on . Also, we have Since is integrable on , from (16) we know that is also integrable on . The above arguments assure that Lemma 4 is applicable, then applying Lemma 4 on subharmonic function on we have
Putting the above equation into (15) and noting that the warping function is a smooth positive function on , we obtain , thus, using inequalities (1) and (12) in this equation we have
The hypothesis (1) implies that on , then it follows from the above inequality that ; comparing this inequality with inequality (12) we obtain an identity . Finally, using and in (7) gives that which means that is a constant on . Then we prove that is a slice of .
Next we give the Riemannian version of the proof of Theorem 1 on a Riemannian warped product space as follows.
Proof of Theorem 2. In this context, we may consider being the orientation of the hypersurface such that its angle function satisfies
Now, letting , then it follows from Lemma 3 that
The assumption (2) implies that is positive, then, from (20) we have that
Noticing that the warping function is positive on , then, using the above inequality and (2) in (21) we obtain
this means that is a subharmonic function on .
As the spacelike hypersurface is bounded away from the infinity of , then the height function is also bounded on . Also, (16) holds in this context. As is integrable on , then (16) assures that is also integrable. From the above arguments we see that Lemma 4 is applicable; applying Lemma 4 on subharmonic function we obtain (17). Thus, putting (17) into (23) and noting that the warping function is positive on , we obtain , using inequality (2) in this equation yields that
It follows from (24) that , comparing this inequality with inequality (20) we obtain identity . Finally, using and in (7) gives which means that is a constant on . Then we prove that is a slice of .
Remark 5. It is worth to point out that Colares and de Lima  obtained some rigidity theorems in semi-Riemannian warped products which are similar to ours. However, in the assumptions of Theorems 4.2 and 4.6 of , the warping function is assumed to has convex logarithm. On the other hand, we refer the reader to [12, 15] for some examples of semi-Riemannian warped products whose warping functions are not necessarily to have convex logarithm. That is, without requiring the assumption that is convex, our Theorems 1 and 2 attain the same conclusions as the corresponding results proved in .
In this section, we apply our main theorems on some physical models, in particular, on steady state-type space spacetime and hyperbolic-type space.
According to , is said to be a steady state-type spacetime, where is an -dimensional complete and connected Riemannian manifold. In particular, is called the -dimensional steady state spacetime, which is isometric to an open subset of the de Sitter space . The importance of studying the steady state spacetime comes from the fact that, in cosmology, is the steady state model of the universe proposed by Bondi and Gold , and Hoyle .
Suppose that the warping function is given by , then the following result follows from Theorem 1.
Corollary 6. Let be a complete spacelike hypersurface in steady state-type spacetime. Suppose that is bounded away from the infinity of and that the mean curvature satisfies . If has integrable norm on , then and is a slice of .
The hyperbolic-type space is defined by , where is a complete connected Riemannian manifold. The motivation for investigating the hyperbolic-type space comes from the fact that the -dimensional hyperbolic space is isometric to . Noting that an explicit isometry between the half-space model and this hyperbolic-type model has been pointed out by Alías and Dajczer in .
Now letting the warping function be for , then the following result follows from Theorem 2.
Corollary 7. Let be a complete spacelike hypersurface in hyperbolic-type space. Suppose that is bounded away from the infinity of and that the mean curvature satisfies . If has integrable norm on , then and is a slice of .
Also, letting the warping function be and an -dimensional Euclidean space, then the following result follows from Corollary 7.
Corollary 8. Let be a complete spacelike hypersurface in a hyperbolic space. Suppose that is bounded away from the infinity of and that the mean curvature satisfies . If has integrable norm on , then and is a horosphere of .
Remark 9. De Lima in  proved that a complete spacelike hypersurface immersed in hyperbolic space with the mean curvature and is a horosphere, provided that is under a horosphere of the hyperbolic space and the second fundamental from is bounded.
The project is supported by the Natural Science Foundation of China (no. 10931005) and the Natural Science Foundation of Guangdong Province of China (no. S2011010000471).
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