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
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DOI：
10.4310/cag.2011.v19.n4.a1
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发表时间：
2010-11
期刊：
影响因子：
--
通讯作者：
N. Le;N. Šešum
中科院分区：
文献类型：
--
作者：
N. Le;N. Šešum
在本文中，我们证明了平均曲率爆破在相同的速度作为第二基本形式在第一个奇异时间T$的任何紧凑，I型平均曲率流。对于曲面的平均曲率流，只要高斯密度小于2，我们也得到了类似的结果。我们的证明是基于连续重新标度和分类的自收缩。我们表明，所有的奇异集的概念定义在\cite{St}符合任何类型的I平均曲率流，从而推广了斯通的结果，谁建立了任何平均凸型I平均曲率流。我们还建立了一个间隙定理的自收缩。
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.