The mean curvature at the first singular time of the mean curvature flow

TL;DR

This paper proves that mean curvature necessarily blows up at the first singular time of mean curvature flow under certain conditions, extending understanding of singularity formation in geometric flows.

Contribution

It establishes conditions under which mean curvature blows up at the first singular time, including for all type I singularities and in certain two-dimensional cases.

Findings

Mean curvature blows up at first singular time for type I singularities.

In 2D, mean curvature blows up if the Multiplicity One Conjecture holds or Gaussian density is less than two.

Develops a local regularity theorem analogous to Choi-Schoen estimate.

Abstract

Consider a family of smooth immersions $F(\cdot,t): M^n\to \mathbb{R}^{n+1}$ of closed hypersurfaces in $\mathbb{R}^{n+1}$ moving by the mean curvature flow $\frac{\partial F(p,t)}{\partial t} = -H(p,t)\cdot \nu(p,t)$, for $t\in [0,T)$. We prove that the mean curvature blows up at the first singular time $T$ if all singularities are of type I. In the case $n = 2$, regardless of the type of a possibly forming singularity, we show that at the first singular time the mean curvature necessarily blows up provided that either the Multiplicity One Conjecture holds or the Gaussian density is less than two. We also establish and give several applications of a local regularity theorem which is a parabolic analogue of Choi-Schoen estimate for minimal submanifolds.