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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 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.

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.

We introduce a new approach to the local study of maximal surfaces in Lorentz-Minkowski space, based on a complex representation formula for this kind of surfaces. As an application we solve a certain Björling-type problem in Lorentz-Minkowski space and we obtain some results related to it. We also establish, springing from this complex representation, a way of introducing examples of maximal surfaces with interesting prescribed geometric properties. Further applications of the complex representation let us inspect some known results from a different perspective, and show how our approach can be used to classify certain families of maximal surfaces. † Partially supported by Dirección General de Investigación (MCYT) BFM2001-2871 and Consejería de Educación y Universidades (CARM) Programa Séneca, PI-3/00854/FS/01.

We study hypersurfaces in Euclidean space R n+1 whose position vector x satisfies the condition L k x = Ax + b, where L k is the linearized operator of the (k + 1)th mean curvature of the hypersurface for a fixed k = 0, . . . , n − 1, A ∈ R (n+1)×(n+1) is a constant matrix and b ∈ R n+1 is a constant vector. For every k, we prove that the only hypersurfaces satisfying that condition are hypersurfaces with zero (k + 1)th mean curvature and open pieces of round hyperspheres and generalized right spherical cylinders of the form S m (r) × R n−m , with k + 1 ≤ m ≤ n − 1. This extends a previous classification for hypersurfaces in R n+1 satisfying x = Ax + b, where = L 0 is the Laplacian operator of the hypersurface, given independently by

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