On Ilmanen’s multiplicity-one conjecture for mean curvature flow with type-$I$ mean curvature

Abstract

In this paper, we show that if the mean curvature of a closed smooth embedded mean curvature flow in ${\mathbb{R}}^3$ is of type-$I$, then the rescaled flow at the first finite singular time converges smoothly to a self-shrinker flow with multiplicity one. This result confirms Ilmanen's multiplicity-one conjecture under the assumption that the mean curvature is of type-$I$. As a corollary, we show that the mean curvature at the first singular time of a closed smooth embedded mean curvature flow in ${\mathbb{R}}^3$ is at least of type-$I$.