Sobre self-shrinkers e self-expanders completos

Abstract

In this thesis, we present rigidity results for the sphere, the plane, and the cylinder, as they are the only Self-shrinkers that satisfy an important classical geometric condition. This condition is as follows: the union of all affine and tangent submanifolds to a complete Self-shrinker omits a non-empty set in Euclidean space. This assumption leads us to a new class of submanifolds, distinct from those with polynomial volume growth or those considered proper. Additionally, we also demonstrate a result for self-expanders that involves the same classical geometric condition.