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The extension and convergence of mean curvature flow in higher codimension


Authors: Kefeng Liu, Hongwei Xu, Fei Ye and Entao Zhao
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 53C44, 53C40
DOI: https://doi.org/10.1090/tran/7281
Published electronically: November 1, 2017
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Abstract: In this paper, we investigate the convergence of the mean curvature flow of closed submanifolds in $ \mathbb{R}^{n+q}$. We show that if the initial submanifold satisfies some suitable integral curvature conditions, then along the mean curvature flow it will shrink to a round point in finite time.


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Kefeng Liu
Affiliation: Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China – and – Department of Mathematics, University of California Los Angeles, Box 951555, Los Angeles, California, 90095-1555
Email: kefeng@zju.edu.cn, liu@math.ucla.edu

Hongwei Xu
Affiliation: Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China
Email: xuhw@zju.edu.cn

Fei Ye
Affiliation: Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China
Email: yf@cms.zju.edu.cn, flemmingye@163.com

Entao Zhao
Affiliation: Center of Mathematical Sciences, Zhejiang University, Hangzhou 310027, People’s Republic of China
Email: zhaoet@zju.edu.cn

DOI: https://doi.org/10.1090/tran/7281
Keywords: Mean curvature flow, submanifold, maximal existence time, convergence theorem, integral curvature
Received by editor(s): September 23, 2016
Received by editor(s) in revised form: January 23, 2017
Published electronically: November 1, 2017
Additional Notes: This research was supported by the National Natural Science Foundation of China, Grant Nos. 11531012, 11371315, 11201416.
Article copyright: © Copyright 2017 American Mathematical Society