中曲率流光滑Lipschitz子流形
The Mean Curvature Flow Smoothes Lipschitz Submanifolds
摘要：
中曲率流是子流形空间上体积泛函的梯度流。证明了中曲率流的一个基本正则性结果：局部Lipschitz范数小的Lipschitz子流形沿中曲率流瞬时光滑.这推广了Ecker和Huisken关于Lipschitz超曲面的正则性定理。特别地，欧几里得空间中任何具有连续诱导度量的子流形都可以被中曲率流光滑化。从Lawson和Osserman的例子来看，在高余维情况下，小性假设是必要的。中曲率流的稳定相对应于极小子流形。从而推广了Morrey关于$C^1$极小子流形光滑性的经典定理.
英文摘要：
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.
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