Pure and Applied Mathematics Quarterly

Volume 11 (2015)

Number 2

Contracting pinched hypersurfaces in spheres by their mean curvature

Pages: 329 – 368

DOI: https://dx.doi.org/10.4310/PAMQ.2015.v11.n2.a8

Authors

Yan Li (Center of Mathematical Sciences, Zhejiang University, Zhejiang, Hangzhou, China; and School of Mathematical Science, Peking University, Beijing, China)

Hongwei Xu (Center of Mathematical Sciences, Zhejiang University, Zhejiang, Hangzhou, China)

Entao Zhao (Center of Mathematical Sciences, Zhejiang University, Zhejiang, Hangzhou, China)

Abstract

In this paper, we study an open problem proposed in [10]. We prove that the mean curvature flow of hypersurfaces in the sphere will contract to a round point in finite time if the initial hypersurface satisfies a curvature pinching condition. Our theorem is a partial improvement of the convergence theorem due to Huisken [7].

Keywords

mean curvature flow, hypersurface, sphere, curvature pinching

2010 Mathematics Subject Classification

53C40, 53C44

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Published 24 August 2016