Márcio Rostirolla Adames

Spacelike Self-similar Solutions of the Mean Curvature Flow: in Pseudo-euclidean Spaces Márcio Rostirolla Adames

The Mean Curvature Flow is, maybe, the most natural way to deform an immersed submanifold; it deforms an immersion into something "rounder" or "more regular". The Mean Curvature Flow is a much studied tool and one of its problems is that it also produces singularities. These singularities are related to some kinds of self-similar solutions of the MCF. A very important class of self-similar solutions is formed by the self-shrinkers. These are homotheties generated by the MCF which shrink the initial immersion. There are several works about singularity formation for the MCF in Euclidean Space (specially in lower dimension and codimension 1) and special interest into classifying these self-shrinkers because of their relation to the singularities of the MCF. In this book the autor studies the self-shrinkers of the MCF with higher codimension in Pseudo-Euclidean space. The results in this book generalize results of Smoczyk and Huisken, beyond this the non-existence of such self-shrinkers is proven in several cases.