Hipersuperfícies tipo-espaço com curvatura de ordem superior constante

Abstract

In this Thesis, our objects of study are the space-like hypersurfaces immersed in a space-time with some higher order curvature constant. In relation to these hypersurfaces, we address issues such as existence, uniqueness and estimation of geometric quantities associated with thesame. More precisely, we developed Minkowski-type formulas for compact space-like hypersurfaces with edges immersed in the space of De Sitter and having some constant higher order curvature. Then, we apply these formulas to establish a relationship between the curvature mean and the geometry of the edge. We also obtain a sharp estimate for the height function of compact space-like hypersurfaces immersed in the Lorentz-Minkowski L space n+1 with some non-zero constant higher order curvature. As an application of this estimate, we deal with the nature of an end of a complete space-like hypersurface of Ln+1. Finally, we study the existence and uniqueness of complete vertical graphs with constant mean curvature in both the Steady State space Hn+1 and in hyperbolic space Hn+1. As a consequence of this study, we obtain Bernstein-type results in H3 and H3.