Fluxo de curvatura média e self-shrinkers

Abstract

Mean curvature flow is a geometric flow that rises when evolving a hipersurface in the direction of its normal vector field with velocity at each point equal to the mean curvature at the very same point. As it is an evolution flow, a natural question to ask is if it becomes singular (either by losing smoothness or by degenerating). We will see that this is no rare case, so another question that we might ask is regarding the asymptotic behaviour of this evolution or, in other words, what are the hypersurfaces "shapes" when their flow approaches being singular. In his paper of 1990 "Asymptotic behaviour for singularities of the mean curvature flow", Gerard Huisken proved his monotonicity formula and, through that, under certain hypothesis, Huisken concluded that the Self Shrinkers are the hypersurfaces that models the asymptotic behaviour of the flow. This work will discuss all these questions, prove the existence and uniqueness of the mean curvature flow and will also classify the self shrinkers embedded in R3 that have a constant norm of second fundamental form.