The Study Of Spacelike Hypersurfaces In Semi-Riemannian Warped Products

In1915, S. Bernstein have proved the following celebrated theorem:the entire min-imal graph in a three dimensional Euclidean space R3must be a plane. Later, in1970, the analogy of the Bernstein’s theorem in Lorentz-Minkowski spacetime Ln (n≤4) was proved by E. Calabi. The generalizations of the above two results on Euclidean space Rn and Lorentz-Minkowski spacetime Ln of higher dimension are said to be Bernstein-type problem. Historically, many researchers at home and abroad have studied this problems and achieved many important results. In the last two decades, because of some important results obtained by A. J. Alias and S. Montiel, Bernstein-type problems have drawn more and more geometers’s interests and were studied by them. In this dissertation, we shall investigate the Bernstein-type problem in semi-Riemannian warped products. By using the result regarding the harmonic functions on complete Riemannian manifolds due to S. Y. Yau, we prove some Bernstein-type theorems and extend some known results. This dissertation is organized as follows:Chapter1is the Introduction, after introducing the background of the Bernstein-type problem in Euclidean space Rn and Lorentz-Minkowski spacetime Ln, we present a brief outline of many recent results in this framework. At last, we briefly introduce our main results which generalize some known theorems.Chapter2contains many preliminary knowledge, we give some fundamental formulas and properties of Riemannian manifolds in the first section and geometry of submanifolds in semi-Riemannian manifolds in the second section. In the last section, we present some necessarily knowledge of semi-Riemannian warped products and the spacelike hypersur-faces immersed in warped products.Chapter3is the core of this dissertation. We first recall several important lemmas which were used to prove our main theorems later, and then we first give a Bernstein-type result for spacelike hypersurfaces immersed in generalized Robertson-Walker spacetimes. That is, suppose∑n is a complete and connected spacelike hypersurfaces in a generalized Robertson-Walker spacetime with its mean curvature H satisfying if▽h has integrable norm on∑n, then∑n must be a slice. The Riemannian version of the above theorem can be presented as follows:let En be a complete and connected spacelike hypersurfaces in a Riemannian warped product with its mean curvature H satisfying if▽h has integrable norm on∑n, then∑n must be a slice. Finally, in the lase section, we give a summary of results shown in this chapter, pointing our the differences and the improvements with others works.Chapter4is devoted to studying the applications of our main results presented in the previous Chapter in some important mathematical and physical models. We consider complete spacelike hypersurfaces in steady state-type and hyperbolic-type spaces respec-tively, giving some geometric conditions which make these hypersurfaces being slices of those models. As an application of our main results, the sufficient condition for an entire vertical graph in generalized Robertson-Walker spacetime to be a slice was also given.