Caracterização de hipersuperfícies tipo-espaço completas com curvatura média constante satisfazendo uma desigualdade do tipo Okumura.

Abstract

In this work, we state and demonstrate two results for the characterization of complete space-like hypersurfaces immersed, with constant mean curvature, in a Lorentzian shape space satisfying an Okumura-type inequality. More specifically in the first theorem we work with the de Sitter space while the second theorem deals with Lorentzian shape spaces, that is, the Lorentz-Minkowski, de Sitter and Anti-de Sitter spaces. In both results we use an Okumuratype inequality to establish relations between the umbilicity operator and the mean curvature in order to establish that the only space-like hypersurfaces satisfying certain previously established conditions are some hyperbolic cylinders.