Paper · Mar 1, 2025

A gap Theorem on closed self-shrinkers of mean curvature flow

Yuhang Zhao

math.DGWider flowsarXiv:2503.00505v2
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Abstract

In this paper, we prove a pinching theorem for dimensional closed self-shrinkers of the mean curvature flow. If the squared norm of the second fundamental form of a closed self-shrinker of arbitrary codimension satisfies: | \vec{\uppercase\expandafter{\romannumeral2}} |^2 \le 1 +\frac{1}{10 π(n+2)}, then it must be the standard sphere . This result may provide some evidence for the open problem 13.76 in.

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Submitted
Mar 1, 2025
Last revised
Mar 17, 2025 (v2)
Subjects
math.DG

Comments: 18 pages

Mean curvature flowSelf-shrinkers & translatorsCurvature pinching

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 2021, 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 Closed embedded self-shrinkers of mean curvature flow (Oskar Riedler, 2022)5Riedler 2022 Gap theorems for complete self-shrinkers of r-mean curvature flows (Hilário Alencar, G. Pacelli Bessa and Gregório Silva Neto, 2024)2Alencar 2024 Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere (Artemis A. Vogiatzi, 2024)7Vogiatzi 2024 Classification of quadratically pinched self-shrinkers in higher codimension (Debora Impera, Michele Rimoldi and Francesco Ruatta, 2026)3Impera 2026 A mean curvature flow approach to Hamilton's pinching theorem (Liang Cheng and Zhenyu Lu, 2026)4Cheng 2026 The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow (Fagui Li and Yuhang Zhao, 2026)1Li 2026 The rigidity theorems for complete self-expander of mean curvature flow (Zhi Li and Guoxin Wei, 2026)6Li 2026 The planarity estimate in dimensions five and six (Stephen Lynch, 2026)8Lynch 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow

    Fagui Li and Yuhang ZhaoWider flows

    Topics Curvature pinching, Self-shrinkers & translators · Same author Yuhang Zhao · Shared terms “pinching theorem”

  2. Gap theorems for complete self-shrinkers of -mean curvature flows

    Hilário Alencar, G. Pacelli Bessa and Gregório Silva NetoWider flows

    Topics Self-shrinkers & translators · Shared terms “gap theorem”, “fundamental form”

  3. Classification of quadratically pinched self-shrinkers in higher codimension

    Debora Impera, Michele Rimoldi and Francesco RuattaWider flows

    Topics Curvature pinching, Self-shrinkers & translators · Shared terms “Andrews”, “fundamental form”

  4. A mean curvature flow approach to Hamilton's pinching theorem

    Liang Cheng and Zhenyu LuWider flows

    Topics Curvature pinching · Shared terms “pinching theorem”, “extrinsic”

  5. Closed embedded self-shrinkers of mean curvature flow

    Oskar RiedlerWider flows

    Topics Self-shrinkers & translators · Shared terms “closed self”

  6. The rigidity theorems for complete self-expander of mean curvature flow

    Zhi Li and Guoxin WeiWider flows

    Topics Curvature pinching, Self-shrinkers & translators · Shared terms “fundamental form”

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

    Artemis A. VogiatziWider flows

    Topics Curvature pinching, Self-shrinkers & translators

  8. The planarity estimate in dimensions five and six

    Stephen LynchWider flows

    Topics Curvature pinching · Shared terms “Andrews”, “fundamental form”

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) 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)

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.