Paper · Mar 13, 2021

A sharp convergence theorem for the mean curvature flow in spheres I

Dong Pu

math.DGWider flowsarXiv:2103.07702
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Abstract

In this paper, we prove a sharp convergence theorem for the mean curvature flow of arbitrary codimension in spheres which improves Baker's convergence theorem. In particular, we obtain a new differentiable sphere theorem for submanifolds in spheres.

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Submitted
Mar 13, 2021
Subjects
math.DG

Comments: 21 pages

Mean curvature flowSphere theorems

Its neighbourhood

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Mean curvature flow of arbitrary codimension in complex projective spaces (Li Lei and Hongwei Xu, 2016)3Lei 2016 Some sharp differential sphere theorems for nonnegative scalar curvature manifolds (Qing Cui and Linlin Sun, 2017)6Cui 2017 The mean curvature flow by parallel hypersurfaces (Hiuri Fellipe Santos dos Reis and Keti Tenenblat, 2017)5dos Reis 2017 A conformally invariant gap theorem characterizing ℂP}² via the Ricci flow (Sun-Yung A. Chang, Matthew Gursky and Siyi Zhang, 2018)8Chang 2018 Ancient Solution of Mean Curvature Flow in Space Forms (Li Lei, Hongwei Xu and Entao Zhao, 2019)7Lei 2019 New Developments in Mean Curvature Flow of Arbitrary Codimension Inspired By Yau Rigidity Theory (Li Lei and Hong-Wei Xu, 2020)1Lei 2020 Surfaces pinched by normal curvature for mean curvature flow in space forms (Dong Pu, Jingjing Su and Hongwei Xu, 2020)2Pu 2020 Rigidity of area non-increasing maps (Man-Chun Lee, Luen-Fai Tam and Jingbo Wan, 2023)4Lee 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. New Developments in Mean Curvature Flow of Arbitrary Codimension Inspired By Yau Rigidity Theory

    Li Lei and Hong-Wei XuWider flows

    Topics Sphere theorems · Shared terms “convergence theorem”, “differentiable sphere”

  2. Surfaces pinched by normal curvature for mean curvature flow in space forms

    Dong Pu, Jingjing Su and Hongwei XuWider flows

    Topics Sphere theorems · Same author Dong Pu · Shared terms “convergence theorem”, “Baker”

  3. Mean curvature flow of arbitrary codimension in complex projective spaces

    Li Lei and Hongwei XuWider flows

    Topics Sphere theorems · Shared terms “convergence theorem”, “differentiable sphere”

  4. Rigidity of area non-increasing maps

    Man-Chun Lee, Luen-Fai Tam and Jingbo Wan

    Topics Sphere theorems

  5. The mean curvature flow by parallel hypersurfaces

    Hiuri Fellipe Santos dos Reis and Keti TenenblatWider flows

    Topics Sphere theorems

  6. Some sharp differential sphere theorems for nonnegative scalar curvature manifolds

    Qing Cui and Linlin Sun

    Topics Sphere theorems

  7. Ancient Solution of Mean Curvature Flow in Space Forms

    Li Lei, Hongwei Xu and Entao ZhaoWider flows

    Topics Sphere theorems

  8. A conformally invariant gap theorem characterizing via the Ricci flow

    Sun-Yung A. Chang, Matthew Gursky and Siyi Zhang

    Topics Sphere theorems

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.