Abstract
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.
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22 May 2021
A Correction to this paper has been published: https://doi.org/10.1007/s12220-021-00692-3
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Acknowledgements
The author want to thank Professor Bill Minicozzi for his helpful comments. The author is also grateful to Zhichao Wang, Jinxin Xue and Xin Zhou for the invaluable discussions. Finally, the author thanks the anonymous referees for the comments and suggestions.
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Sun, A. Entropy in a Closed Manifold and Partial Regularity of Mean Curvature Flow Limit of Surfaces. J Geom Anal 31, 5619–5635 (2021). https://doi.org/10.1007/s12220-020-00494-z
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DOI: https://doi.org/10.1007/s12220-020-00494-z