Uniqueness and nonexistence of complete spacelike hypersurfaces, Calabi-Bernstein type results and applications to Einstein-de Sitter and steady state type spacetimes

We investigate the geometry of complete spacelike hypersurfaces (immersed) in a generalized Robertson-Walker spacetime - I xf M-n. Under suitable constraints on the sectional curvature of the Riemannian fiber M-n, on the warping function f and on the future mean curvature (that is, the mean curvature function with respect to the future-pointing Gauss map of the spacelike hypersurface), we are able to prove that such a spacelike hypersurface must be a slice {t} x M-n of the ambient spacetime. Nonexistence and Calabi-Bernstein type results concerning entire spacelike graphs constructed over the Riemannian fiber M-n are also obtained, as well as applications to the Einstein-de Sitter and steady state type spacetimes are given. Our approach is based on the so-called Omori-Yau's generalized maximum principle and on certain integrability properties due to Yau.