On Complete Spacelike Hypersurfaces in a Semi-Riemannian Warped Product

Yaning Wang,Ximin Liu

doi:10.1155/2013/757041

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.

Highlights

The theory of spacelike hypersurfaces immersed in semiRiemannian warped products with constant mean curvature has got increasing interest both from geometers and physicists recently
The aim of this paper is to study such type problems
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

Summary

Introduction

The theory of spacelike hypersurfaces immersed in semiRiemannian warped products with constant mean curvature has got increasing interest both from geometers and physicists recently. 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:. Replacing the null convergence condition by (ln f)󸀠󸀠 ≤ 0, Alıas 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. Suppose that Σn is bounded away from the infinity of Mn+1 and that sup p∈Σn f󸀠 f (6)

Preliminaries

Key Lemmas

Proofs of Main Theorems

Application of Main Theorems

Entire Vertical Graphs in Robertson-Walker Spacetime