A. Gervasio Colares

Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes

In this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson�Walker (GRW) spacetimes. In particular, we consider the following question: under what conditions must a compact spacelike hypersurface with constant higher order mean curvature in a spatially closed GRW spacetime be a spacelike slice? We prove that this happens, essentially, under the so called null convergence condition. Our approach is based on the use of the Newton transformations (and their associated differential operators) and the Minkowski formulae for spacelike hypersurfaces.

INTEGRAL FORMULAE FOR SPACELIKE HYPERSURFACES IN CONFORMALLY STATIONARY SPACETIMES AND APPLICATIONS

In this paper we develop general Minkowski-type formulae for compact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds admitting a timelike conformal field. We apply them to the study of the umbilicity of compact spacelike hypersurfaces in terms of their r-mean curvatures. We derive several uniqueness results, for instance, compact spacelike hypersurfaces are umbilical if either some of their r-mean curvatures are linearly related or one of them is constant.

A gap theorem for complete constant scalar curvature hypersurfaces in the de Sitter space

To each immersed complete space-like hypersurfaceM with constant normalized scalar curvature R in the de Sitter space S nC1 1 , we associate sup H 2 , where H is the mean curvature of M .I t is proved that the condition sup H 2 Cn. N R/, where N R D .R 1 /> 0 and Cn. N R/ is a constant depending only on R and n, implies that either M is totally umbilical or M is a hyperbolic cylinder. It is also proved the sharpness of this result by showing the existence of a class of new rotation constant scalar curvature hypersurfaces in S nC1 1 such that sup H 2 >C n. N R/.

On the rigidity of spacelike hypersurfaces immersed in the steady state space H elevado a n+1

In this paper, as a suitable application of the well known generalized Maximum Principle of Omori{Yau, we obtain rigidity results concerning to complete spacelike hypersurfaces immersed in the half Hn+1 of the de Sitter space Sn+1 1 , which models the so-called steady state space. Moreover, by using an isometrically equivalent model for Hn+1, we extend our results to a wider family of spacetimes. Finally, we also study the uniqueness of entire vertical graphs in such ambient spacetimes.

A Further Characterization Of Ellipsoids

In this note we extend a characterization of ellipsoids given in [6, Theorem 3.1.2.7], which is related to the classical characterization of ellipsoids as the only ovaloids with constant r-th affine mean curvature.