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Transactions of the American Mathematical Society

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Ancient solution of mean curvature flow in space forms


Authors: Li Lei, Hongwei Xu and Entao Zhao
Journal: Trans. Amer. Math. Soc. 374 (2021), 2359-2381
MSC (2020): Primary 53C40
DOI: https://doi.org/10.1090/tran/8267
Published electronically: January 27, 2021
MathSciNet review: 4223019
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Abstract: In this paper we investigate the rigidity of ancient solutions of the mean curvature flow with arbitrary codimension in space forms. We first prove that under certain sharp asymptotic pointwise curvature pinching condition the ancient solution in a sphere is either a shrinking spherical cap or a totally geodesic sphere. Then we show that under certain pointwise curvature pinching condition the ancient solution in a hyperbolic space is a family of shrinking spheres. We also obtain a rigidity result for ancient solutions in a nonnegatively curved space form under an asymptotic integral curvature pinching condition.


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Additional Information

Li Lei
Affiliation: Center of Mathematical Sciences, Zhejiang University, Zhejiang Hangzhou, 310027, People’s Republic of China
ORCID: 0000-0002-7143-8101
Email: lei-li@zju.edu.cn

Hongwei Xu
Affiliation: Center of Mathematical Sciences, Zhejiang University, Zhejiang Hangzhou, 310027, People’s Republic of China
MR Author ID: 245171
Email: xuhw@zju.edu.cn

Entao Zhao
Affiliation: Center of Mathematical Sciences, Zhejiang University, Zhejiang Hangzhou, 310027, People’s Republic of China
MR Author ID: 884026
Email: zhaoet@zju.edu.cn

Keywords: Ancient solution, mean curvature flow of submanifolds, pinching theorem, second fundamental form
Received by editor(s): December 12, 2019
Published electronically: January 27, 2021
Additional Notes: This research was supported by the National Natural Science Foundation of China, Grant Nos. 11531012, 12071424; and the China Postdoctoral Science Foundation, Grant No. BX20180274.
Dedicated: Dedicated to Professor Katsuhiro Shiohama on the occasion of his 80th birthday
Article copyright: © Copyright 2021 American Mathematical Society