In this paper we establish new Calabi-Bernstein results for maximal surfaces immersed into a Lorentzian product space of the form M 2 × R 1 , where M 2 is a connected Riemannian surface and M 2 × R 1 is endowed with the Lorentzian metric , = , M − dt 2 . In particular, when M is a Riemannian surface with non-negative Gaussian curvature K M , we prove that any complete maximal surface in M 2 × R 1 must be totally geodesic. Besides, if M is non-flat we conclude that it must be a slice M × {t 0 }, t 0 ∈ R (here by complete it is meant, as usual, that the induced Riemannian metric on the maximal surface from the ambient Lorentzian metric is complete). We prove that the same happens if the maximal surface is complete with respect to the metric induced from the Riemannian product M 2 × R. This allows us to give also a non-parametric version of the Calabi-Bernstein theorem for entire maximal graphs in M 2 × R 1 , under the same assumptions on K M . Moreover, we also construct counterexamples which show that our Calabi-Bernstein results are no longer true without the hypothesis K M ≥ 0. These examples are constructed via a duality result between minimal and maximal graphs.

Abstract. We consider complete spacelike hypersurfaces with constant mean curvature in the open region of de Sitter space known as the steady state space. We prove that if the hypersurface is bounded away from the infinity of the ambient space, then the mean curvature must be H = 1. Moreover, in the 2-dimensional case we obtain that the only complete spacelike surfaces with constant mean curvature which are bounded away from the infinity are the totally umbilical flat surfaces. We also derive some other consequences for hypersurfaces which are bounded away from the future infinity. Finally, using an isometrically equivalent model for the steady state space, we extend our results to a wider family of spacetimes.

In this paper we establish some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space of the form M 2 × R 1 , where M 2 is a connected Riemannian surface with non-negative Gaussian curvature and M 2 × R 1 is endowed with the Lorentzian product metric , = , M − dt 2 . In particular, and as an application of our main result, we deduce that every maximal graph over a starlike domain ⊆ M is parabolic. This allows us to give an alternative proof of the non-parametric version of the Calabi-Bernstein result for entire maximal graphs in M 2 × R 1 .

In this paper, as a suitable application of the well-known generalized maximum principle of Omori-Yau, we obtain uniqueness results concerning to complete spacelike hypersurfaces with constant mean curvature immersed in a Robertson-Walker (RW) spacetime. As an application of such uniqueness results for the case of vertical graphs in a RW spacetime, we also get non-parametric rigidity results.

Our purpose in this paper is to apply some maximum principles in order to study the rigidity of complete spacelike hypersurfaces immersed in a spatially weighted generalized Robertson-Walker (GRW) spacetime, which is supposed to obey the so called strong null convergence condition. Under natural constraints on the weight function and on the f -mean curvature, we establish sufficient conditions to guarantee that such a hypersurface must be a slice of the ambient space. In this setting, we also obtain new Calabi-Bernstein type results concerning entire graphs in a spatially weighted GRW spacetime.2010 Mathematics Subject Classification. Primary 53C42, 53A07; Secondary 35P15.