Um teorema de rigidez para hipersuperfícies cmc completas em variedades de Lorentz

Abstract

Our aim in this work is to show a classification theorem for complete CMC hipersurfaces in Lorentz manifolds of constant sectional curvature, under certains bounds on the scalar curvature. To this end we use Simons formula, wich gives a relation between Newton transformations and the Laplacian of the squared norm of the Weingarten operator A, as well as a maximum principle due to H. Omori and S. T. Yau. We obtain, as a first application, a classification of complete spacelike CMC hypersurfaces of the De Sitter space, having scalar curvature R maior ou igual a 1. We also conclude that all complete spacelike hypersurfaces with positive constant mean curvature and nonegative scalar curvature in the Lorentz-Minkowski space are cylinders over a plane curve and, up to isometries, we determine this curve.