%0 Journal Article
%T On Bernstein-Type Theorems in Semi-Riemannian Warped Products
%A Wenjie Wang
%A Ximin Liu
%J Advances in Mathematical Physics
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/959143
%X 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). 1. Introduction 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 [1] obtained some unique results concerning complete spacelike hypersurfaces with constant mean curvature immersed in steady state space and hyperbolic space, respectively. Later, Albujer and Al¨ªas [2] 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 [5]. Also, by using the well known result according to Yau [6], Camargo et al. in [7] 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 [8] 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 [6] 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
%U http://www.hindawi.com/journals/amp/2013/959143/