Blow-up rate of the mean curvature during the mean curvature flow and a gap theorem for self-shrinkers

TL;DR

This paper investigates the blow-up behavior of mean curvature during singularities in mean curvature flow, establishing that mean curvature and second fundamental form blow up at the same rate, and introduces a gap theorem for self-shrinkers.

Contribution

It proves the rate at which mean curvature blows up matches that of the second fundamental form at singularities and generalizes singular set coincidence results for Type I flows.

Findings

Mean curvature blows up at the same rate as the second fundamental form.

Singular sets coincide for any Type I mean curvature flow.

A gap theorem for self-shrinkers is established.

Abstract

In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time $T$ of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar result provided that the Gaussian density is less than two. Our proofs are based on continuous rescaling and the classification of self-shrinkers. We show that all notions of singular sets defined in \cite{St} coincide for any Type I mean curvature flow, thus generalizing the result of Stone who established that for any mean convex Type I Mean curvature flow. We also establish a gap theorem for self-shrinkers.