On Complete Spacelike Hypersurfaces in a Semi-Riemannian Warped Product

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.

On Bernstein-Type Theorems in Semi-Riemannian Warped Products

“…Setting = 0 in Lemma 4.1 of [9], 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 [9], the second author and Wang in [10] generalize this result in a semi-Riemannian warped product as follows.…”

“…Lemma 3 (see [9,10]). Let : Σ → × be a spacelike hypersurface immersed in a semi-Riemannian warped product.…”

“…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.…”

Complete spacelike hypersurfaces with positive r-th mean curvature in a semi-Riemannian warped product

“…It is worth to point out that the Laplacian of integral of the warping function was studied by the present authors in [20,21] to obtain some uniqueness results. Throughout this paper, we denote by L(Σ n ) the space of Lebesgue integrable functions on spacelike hypersurface Σ n .…”

On Horizontal Lightlike Hypersurfaces of Robertson-Walker Spacetimes

“…The generalized Robertson-Walker spacetimes are very important models both from mathematical and physical (cosmological) points of view, for more details see Bondi and Gold [3], Hawking and Ellis [8] and Hoyle [9]. Many authors have studied spacelike hypersurfaces of generalized Robertson-Walker spacetimes by requiring certain conditions on the mean curvatures of the hypersurfaces (see [1,4,14]). Recently, degenerate hypersurfaces of generalized Robertson-Walker spacetimes have been investigated by Kang [11].…”