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

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

2010-11-23

arxiv.org/abs/1011.5245v1

Comm. Anal. Geom, 19 (2011), no. 4, 633-659

Mathematics

Differential Geometry

21 pages

Scientific paper

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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