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

### Nam Q. Le

Department of Mathematics, Columbia University, New York, USA### Natasa Sesum

Department of Mathematics, University of Pennsylvania, Philadelphia, PA, USA

## Abstract

Consider a family of smooth immersions $F(⋅,t):M_{n}→R_{n+1}$ of closed hypersurfaces in $R_{n+1}$ moving by the mean curvature flow $∂t∂F(p,t) =−H(p,t)⋅ν(p,t)$, for $t∈[0,T)$. 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.

## Cite this article

Nam Q. Le, Natasa Sesum, The mean curvature at the first singular time of the mean curvature flow. Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 6, pp. 1441–1459

DOI 10.1016/J.ANIHPC.2010.09.002