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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$\frac{\partial Fp,t}{\partial t} = -Hp,t\cdot up,t$, for $t\in 0,T$.We prove that the mean curvature blows up at the first singular time $T$ if allsingularities are of type I. In the case $n = 2$, regardless of the type of apossibly forming singularity, we show that at the first singular time the meancurvature necessarily blows up provided that either the Multiplicity OneConjecture holds or the Gaussian density is less than two. We also establishand give several applications of a local regularity theorem which is aparabolic analogue of Choi-Schoen estimate for minimal submanifolds.



Autor: Nam Le, Natasa Sesum

Fuente: https://arxiv.org/







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