Paper · May 25, 2016

Mean curvature flow of arbitrary codimension in complex projective spaces

Li Lei, Hongwei Xu

math.DGWider flowsarXiv:1605.07963
View PDF View HTML arXiv page

As on arXiv

Abstract

In this paper, we investigate the mean curvature flow of submanifolds of arbitrary codimension in . We prove that if the initial submanifold satisfies a pinching condition, then the mean curvature flow converges to a round point in finite time, or converges to a totally geodesic submanifold as . Consequently, we obtain a new differentiable sphere theorem for submanifolds in . Our work improves the convergence theorem for mean curvature flow due to Pipoli and Sinestrari .

View PDF View HTML arXiv page
Submitted
May 25, 2016
Subjects
math.DG

Comments: 31 pages

Mean curvature flowSingularitiesCurvature pinchingSphere theorems

Its neighbourhood

Connected papers

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 Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes (Giuseppe Pipoli and Carlo Sinestrari, 2016)7Pipoli 2016 Mean curvature flow for pinched submanifolds in rank one symmetric spaces (Naoyuki Koike, Yoshiyuki Mizumura and Nana Uenoyama, 2017)4Koike 2017 Ancient Solution of Mean Curvature Flow in Space Forms (Li Lei, Hongwei Xu and Entao Zhao, 2019)1Lei 2019 New Developments in Mean Curvature Flow of Arbitrary Codimension Inspired By Yau Rigidity Theory (Li Lei and Hong-Wei Xu, 2020)2Lei 2020 Surfaces pinched by normal curvature for mean curvature flow in space forms (Dong Pu, Jingjing Su and Hongwei Xu, 2020)3Pu 2020 A sharp convergence theorem for the mean curvature flow in spheres I (Dong Pu, 2021)5Pu 2021 High Codimension Mean Curvature Flow in the Complex Projective Space (Artemis A. Vogiatzi, 2023)6Vogiatzi 2023 Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere (Artemis A. Vogiatzi, 2024)8Vogiatzi 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ancient Solution of Mean Curvature Flow in Space Forms

    Li Lei, Hongwei Xu and Entao ZhaoWider flows

    Topics Curvature pinching, Sphere theorems · Same authors Li Lei, Hongwei Xu · Shared terms “pinching condition”, “totally geodesic”

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

    Li Lei and Hong-Wei XuWider flows

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

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

    Dong Pu, Jingjing Su and Hongwei XuWider flows

    Topics Curvature pinching, Sphere theorems · Same author Hongwei Xu · Shared terms “convergence theorem”, “pinching condition”

  4. Mean curvature flow for pinched submanifolds in rank one symmetric spaces

    Naoyuki Koike, Yoshiyuki Mizumura and Nana UenoyamaWider flows

    Topics Curvature pinching, Singularities · Shared terms “complex projective”, “projective space”, “geodesic submanifold”

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

    Dong PuWider flows

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

  6. High Codimension Mean Curvature Flow in the Complex Projective Space

    Artemis A. VogiatziWider flows

    Topics Curvature pinching, Singularities · Shared terms “complex projective”, “projective space”, “totally geodesic”

  7. Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes

    Giuseppe Pipoli and Carlo SinestrariWider flows

    Topics Curvature pinching, Singularities · Shared terms “projective space”

  8. Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere

    Artemis A. VogiatziWider flows

    Topics Curvature pinching, Singularities · Shared terms “round point”, “totally geodesic”

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Curvature pinching Curvature conditions that Ricci flow preserves and improves. Hamilton–Ivey pinching makes positive curvature dominate in 3D blow-ups, and invariant… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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. 12B.6
    Bounded Curvature at Bounded DistanceBook 12B · κ-Solutions and Surgery

    The heart of the canonical neighbourhood proof: why curvature at bounded rescaled distance from a point of high curvature stays bounded, via incomplete limits, cones and Hamilton–Ivey pinching (Morgan–Tian, chapter 10).

  2. 6A.9
    Reaction–Diffusion: Fronts, Patterns and PinchingBook 6A · The Heat Equation and Its Relatives

    Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.

  3. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.