The Mean Curvature Flow Smoothes Lipschitz submanifolds

Mu-Tao Wang Columbia University

Differential Geometry mathscidoc:1608.10061

Communications in Analysis and Geometry , 12, (3), 581-599, 2004
The mean curvature flow is the gradient flow of volume functionals on the space of submanifolds. We prove a fundamental regularity result of the mean curvature flow in this paper: a Lipschitz submanifold with small local Lipschitz norm becomes smooth instantly along the mean curvature flow. This generalizes the regularity theorem of Ecker and Huisken for Lipschitz hypersurfaces. In particular, any submanifold of the Euclidean space with a continuous induced metric can be smoothed out by the mean curvature flow. The smallness assumption is necessary in the higher codimension case in view of an example of Lawson and Osserman. The stationary phase of the mean curvature flow corresponds to minimal submanifolds. Our result thus generalizes Morrey’s classical theorem on the smoothness of $C_1$ minimal submanifolds.
Mean Curvature Flow, Lipschitz submanifold
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@inproceedings{mu-tao2004the,
  title={The Mean Curvature Flow Smoothes Lipschitz submanifolds},
  author={Mu-Tao Wang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160821235701486094375},
  booktitle={Communications in Analysis and Geometry },
  volume={12},
  number={3},
  pages={581-599},
  year={2004},
}
Mu-Tao Wang. The Mean Curvature Flow Smoothes Lipschitz submanifolds. 2004. Vol. 12. In Communications in Analysis and Geometry . pp.581-599. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160821235701486094375.
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