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Entropy in a Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces
The Journal of Geometric Analysis ( IF 1.1 ) Pub Date : 2020-08-11 , DOI: 10.1007/s12220-020-00494-z
Ao Sun

Inspired by the idea of Colding and Minicozzi (Ann Math 182:755–833, 2015), we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a closed ambient manifold with non-negative Ricci curvature. Moreover, this entropy is monotone along the mean curvature flow in a closed Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature. As an application, we show the partial regularity of the limit of mean curvature flow of surfaces in a three dimensional Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature.



中文翻译:

闭合流形中的熵和表面平均曲率流极限的部分规则

受到Colding and Minicozzi(Ann Math 182:755–833,2015)的启发,我们定义了一般环境黎曼流形中子流形的(平均曲率流)熵。特别地,该熵等于具有非负Ricci曲率的封闭环境歧管中封闭子流形的面积增长。此外,该熵在具有非负截面曲率和平行Ricci曲率的闭合黎曼流形中沿着平均曲率流是单调的。作为一种应用,我们显示了具有非负截面曲率和平行Ricci曲率的三维黎曼流形中表面平均曲率流极限的局部规律。

更新日期:2020-08-12
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