Communications in Analysis and Geometry

Volume 19 (2011)

Number 4

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

Pages: 633 – 659

DOI: http://dx.doi.org/10.4310/CAG.2011.v19.n4.a1

Authors

Nam Q. Le (Department of Mathematics, Columbia University)

Natasa Sesum (Department of Mathematics, Rutgers University)

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.

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