Local Entropy and Generic Multiplicity One Singularities of Mean Curvature Flow of Surfaces

Abstract

In this paper we prove that the generic singularities of mean curvature flow of closed embedded surfaces in ${\mathbb{R}}^3$ modeled by closed self-shrinkers with multiplicity has multiplicity one. Together with the previous result by Colding-Minicozzi in [CM12], we conclude that the only generic singularity of mean curvature flow of closed embedded surfaces in ${\mathbb{R}}^3$ modeled by closed self-shrinkers is a multiplicity one sphere. We also construct particular perturbations of the flow to avoid those singularities with multiplicity higher than one. Our result partially addresses the well-known multiplicity one conjecture by Ilmanen.

26 pages. Correct some typos and revise the paper according to the suggestions by the referees. Accepted by Journal of Differential Geometry