Paper · Apr 17, 2026

A mean curvature flow approach to Hamilton's pinching theorem

Liang Cheng, Zhenyu Lu

math.DGWider flowsarXiv:2604.15792v2
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Abstract

In this paper, we provide a proof of Hamilton's extrinsic pinching theorem using the mean curvature flow approach.

Mean curvature flowCurvature 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 Three-manifolds with non-negatively pinched Ricci curvature (Man-Chun Lee and Peter M. Topping, 2022)8Lee 2022 Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds (Artemis A. Vogiatzi and Huy T. Nguyen, 2023)6Vogiatzi 2023 On locally conformally flat manifolds with positive pinched Ricci curvature (Liang Cheng, 2023)1Cheng 2023 On mean curvature flow solitons in the sphere (Marco Magliaro, Luciano Mari, Fernanda Roing et al., 2025)7Magliaro 2025 A gap Theorem on closed self-shrinkers of mean curvature flow (Yuhang Zhao, 2025)3Zhao 2025 The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow (Fagui Li and Yuhang Zhao, 2026)4Li 2026 Local Convexity and Planarity Estimates for Mean Curvature Flow (Liang Cheng, 2026)2Cheng 2026 Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow (Tsz-Kiu Aaron Chow, 2026)5Chow 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On locally conformally flat manifolds with positive pinched Ricci curvature

    Liang ChengWider flows PDF

    Topics Curvature pinching · Same author Liang Cheng · Shared terms “Hamilton pinching”

  2. Local Convexity and Planarity Estimates for Mean Curvature Flow

    Liang ChengWider flows PDF

    Topics Curvature pinching · Same author Liang Cheng

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

    Yuhang ZhaoWider flows PDF

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

  4. The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow

    Fagui Li and Yuhang ZhaoWider flows PDF

    Topics Curvature pinching · Shared terms “pinching theorem”

  5. Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow

    Tsz-Kiu Aaron Chow PDF

    Topics Curvature pinching

  6. Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds

    Artemis A. Vogiatzi and Huy T. NguyenWider flows PDF

    Topics Curvature pinching

  7. On mean curvature flow solitons in the sphere

    Marco Magliaro, Luciano Mari, Fernanda Roing et al.Wider flows PDF

    Topics Curvature pinching · Shared terms “pinching theorem”

  8. Three-manifolds with non-negatively pinched Ricci curvature

    Man-Chun Lee and Peter M. Topping PDF

    Topics Curvature pinching · Shared terms “Hamilton pinching”

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) 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.