Abstract. In this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized RobertsonWalker (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, esentially, 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.

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.

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Abstract.We obtain some nonexistence results for complete noncompact stable hyppersurfaces with nonnegative constant scalar curvature in Euclidean spaces. As a special case we prove that there is no complete noncompact strongly stable hypersurface M in R 4 with zero scalar curvature S 2 , nonzero GaussKronecker curvature and finite total curvature (i.e. R M |A| 3 < +∞).

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