Paper · Apr 29, 2020

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

Dong Pu, Jingjing Su, Hongwei Xu

math.DGWider flowsarXiv:2004.14258
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Abstract

In this paper, we investigate the mean curvature flow of compact surfaces in -dimensional space forms. We prove the convergence theorems for the mean curvature flow under certain pinching conditions involving the normal curvature, which generalise Baker-Nguyen's convergence theorem.

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Submitted
Apr 29, 2020
Subjects
math.DG
Mean curvature flowCurvature 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 Surfaces of Co-dimension Two Pinched by Normal Curvature (Charles Baker and Huy The Nguyen, 2016)4Baker 2016 Mean curvature flow of arbitrary codimension in complex projective spaces (Li Lei and Hongwei Xu, 2016)3Lei 2016 New pinching estimates for Inverse curvature flows in space forms (Yong Wei, 2017)6Wei 2017 Pinched Ancient Solutions to the High Codimension Mean Curvature Flow (Stephen Lynch and Huy The Nguyen, 2017)7Lynch 2017 A planarity estimate for pinched solutions of mean curvature flow (Keaton Naff, 2019)8Naff 2019 Ancient Solution of Mean Curvature Flow in Space Forms (Li Lei, Hongwei Xu and Entao Zhao, 2019)1Lei 2019 Evolving Pinched Submanifolds of the Sphere by Mean Curvature Flow (Charles Baker and Huy The Nguyen, 2020)5Baker 2020 A sharp convergence theorem for the mean curvature flow in spheres I (Dong Pu, 2021)2Pu 2021
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 author Hongwei Xu · Shared terms “space form”, “pinching condition”

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

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

  4. Surfaces of Co-dimension Two Pinched by Normal Curvature

    Charles Baker and Huy The NguyenWider flows

    Topics Mean curvature flow · Shared terms “normal curvature”

  5. Evolving Pinched Submanifolds of the Sphere by Mean Curvature Flow

    Charles Baker and Huy The NguyenWider flows

    Topics Curvature pinching · Shared terms “normal curvature”, “pinching condition”

  6. New pinching estimates for Inverse curvature flows in space forms

    Yong WeiWider flows

    Topics Curvature pinching, Sphere theorems · Shared terms “space form”

  7. Pinched Ancient Solutions to the High Codimension Mean Curvature Flow

    Stephen Lynch and Huy The NguyenWider flows

    Topics Curvature pinching · Shared terms “pinching condition”, “Baker”, “generalises”

  8. A planarity estimate for pinched solutions of mean curvature flow

    Keaton NaffWider flows

    Topics Curvature pinching · Shared terms “pinching condition”

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) 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. 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.

  2. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.

  3. 11A.10
    Hamilton–Ivey PinchingBook 11A · The Ricci Flow: Existence and Maximum Principles

    Why high curvature in dimension 3 is almost positive.