Uniqueness and pseudolocality theorems of the mean curvature flow
Uniqueness and pseudolocality theorems of the mean curvature flow
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DOI：
10.4310/cag.2007.v15.n3.a1
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发表时间：
2007-03
影响因子：
0.7
通讯作者：
Binglong Chen;L. Yin
中科院分区：
文献类型：
--
作者：
Binglong Chen;L. Yin
Mean curvature flow evolves isometrically immersed base manifolds $M$ in the direction of their mean curvatures in an ambient manifold $\bar{M}$. If the base manifold $M$ is compact, the short time existence and uniqueness of the mean curvature flow are well-known. For complete isometrically immersed submanifolds of arbitrary codimensions, the existence and uniqueness are still unsettled even in the Euclidean space. In this paper, we solve the uniqueness problem affirmatively for the mean curvature flow of general codimensions and general ambient manifolds. In the second part of the paper, inspired by the Ricci flow, we prove a pseudolocality theorem of mean curvature flow. As a consequence, we obtain a strong uniqueness theorem, which removes the assumption on the boundedness of the second fundamental form of the solution.
