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

References in corpus (4)

• Asymptotic structure of self-shrinkers
• The index of shrinkers of the mean curvature flow
• Sharp Entropy Bounds for Plane Curves and Dynamics of the Curve Shortening Flow
• Singularities of Mean Curvature Flow of Surfaces with Additional Forces

Cited by in corpus (6)

• Mean curvature flow with generic initial data
• Initial Perturbation of the Mean Curvature Flow for closed limit shrinker
• On Ilmanen's multiplicity-one conjecture for mean curvature flow with type-I mean curvature
• Lectures on mean curvature flow of surfaces
• On the Entropy of Parabolic Allen-Cahn Equation
• Compactness of self-shrinkers in ${\mathbb{R}}^3$ with fixed genus