Constant Mean Curvature Spacelike Hypersurfaces In Standard Static Spaces: Rigidity And Parabolicity

Our purpose in this paper is investigate the geometry of complete constant mean curvature spacelike hypersurfaces immersed in a standard static space, that is, a Lorentzian manifold endowed with a globally defined timelike Killing vector field. In this setting, supposing that the ambient space is a warped product of the type M-n x rho R-1 whose Riemannian base M-n has nonnegative sectional curvature and the warping function rho is convex on M-n, we use the generalized maximum principle of Omori-Yau in order to establish rigidity results concerning these spacelike hypersurfaces. We also study the parabolicity of maximal spacelike surfaces in M-2 x rho R-1 and we obtain uniqueness results for entire Killing graphs constructed over M-n.