Hipersuperfícies estáveis e fórmula de monotonicidade envolvendo curvatura escalar em espaços de curvatura seccional limitada

Abstract

In this thesis we prove there is no complete hypersurfaces of dimension three in the Euclidean space of dimension four, satisfying some conditions of curvature. Next, we prove in a work jointly with Hilário Alencar, a mean value inequality and a monotonicity formula involving the mean and scalar curvatures of hypersurfaces immersed into a Riemannian manifold of sectional curvature bounded above by some constant. Moreover, we prove a Poincaré type inequality involving the mean curvature and the scalar curvature of hypersurfaces immersed into Riemannian manifolds of sectional curvature bounded above by some constant. We apply this inequality to obtain a result about stable hypersurfaces with zero scalar curvature in Euclidean space.