Teoremas de rigidez para subvariedades do espaço Euclidiano

Abstract

In this thesis we obtained some results of the inexistence and rigidity of submanifolds immersed in the Euclidean space R n+p . In the first part, motivated by the ideas presented in (CAVALCANTE; ESPINAR, 2016), in which the authors extend halfspace type theorems to self-shrinkers, we show that there are no minimal hypersurfaces properly immersed in any region of R n+1 limited by minimal unstable cones. In the second part, we show two results of rigidity that characterize the round spheres as the only closed self-shrinking submanifolds of the Euclidean space, assuming a certain geometric inequality or in the hypothesis that a higher mean curvature is constant. In addition, we have established some correlated results in the context of closed λ-hypersurfaces.