Aspectos geométricos de self-shrinkers de um fluxo geométrico

Abstract

In the first part of this thesis we will study the solutions of the equation Sr = −⟨x, N⟩, that is, the self-shrinkers from the point of view of the theory of submanifolds and for this we will make use of principles of some maximum, namely the Hopf maximum principle and the Omori-Yau maximum principle to obtain rigidity results, classifying the solutions of the aforementioned equation. In the second part we will prove a weighted volume control and obtain an application for the case r = 2, in the third part we will do the classification based on a topological condition, and in the fourth last part we will present non-existence results on warped products.