Paper · Apr 28, 2020

New Developments in Mean Curvature Flow of Arbitrary Codimension Inspired By Yau Rigidity Theory

Li Lei, Hong-Wei Xu

math.DGWider flowsarXiv:2004.13607
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Abstract

In this survey, we will focus on the mean curvature flow theory with sphere theorems, and discuss the recent developments on the convergence theorems for the mean curvature flow of arbitrary codimension inspired by the Yau rigidity theory of submanifolds. Several new differentiable sphere theorems for submanifolds are obtained as consequences of the convergence theorems for the mean curvature flow. It should be emphasized that Theorem 4.1 is an optimal convergence theorem for the mean curvature flow of arbitrary codimension, which implies the first optimal differentiable sphere theorem for submanifolds with positive Ricci curvature. Finally, we present a list of unsolved problems in this area.

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Submitted
Apr 28, 2020
Subjects
math.DG
Journal
Proceedings of the 7th International Congress of Chinese Mathematicians, Vol. I, 327-348, Adv. Lect. Math. 43, Int. Press, Somerville, MA, 2019

Comments: This survey article was published in the Proceedings of the 7th International Congress of Chinese Mathematicians, Vol. I, 327-348, Adv. Lect. Math. 43, Int. Press, Somerville, MA, 2019

Mean curvature flowSphere theoremsStability & uniqueness

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)2Lei 2016 The mean curvature flow by parallel hypersurfaces (Hiuri Fellipe Santos dos Reis and Keti Tenenblat, 2017)7dos Reis 2017 Ancient Solution of Mean Curvature Flow in Space Forms (Li Lei, Hongwei Xu and Entao Zhao, 2019)3Lei 2019 Surfaces pinched by normal curvature for mean curvature flow in space forms (Dong Pu, Jingjing Su and Hongwei Xu, 2020)5Pu 2020 Recent developments in Ricci flows (Richard H Bamler, 2021)8Bamler 2021 A sharp convergence theorem for the mean curvature flow in spheres I (Dong Pu, 2021)1Pu 2021 Ricci flow and PIC1 (Peter M. Topping, 2023)6Topping 2023 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. A sharp convergence theorem for the mean curvature flow in spheres I

    Dong PuWider flows

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

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

    Li Lei and Hongwei XuWider flows

    Topics Sphere theorems · Same author Li Lei · Shared terms “differentiable sphere”, “convergence theorem”

  3. Ancient Solution of Mean Curvature Flow in Space Forms

    Li Lei, Hongwei Xu and Entao ZhaoWider flows

    Topics Sphere theorems, Stability & uniqueness · Same author Li Lei

  4. Rigidity of area non-increasing maps

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

    Topics Sphere theorems, Stability & uniqueness

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

    Dong Pu, Jingjing Su and Hongwei XuWider flows

    Topics Sphere theorems · Shared terms “convergence theorem”

  6. Ricci flow and PIC1

    Peter M. Topping

    Topics Sphere theorems · Shared terms “flow theory”

  7. The mean curvature flow by parallel hypersurfaces

    Hiuri Fellipe Santos dos Reis and Keti TenenblatWider flows

    Topics Sphere theorems

  8. Recent developments in Ricci flows

    Richard H Bamler

    Similar wording in the title and abstract

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.