Paper · Aug 15, 2024

Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere

Artemis A. Vogiatzi

math.DGWider flowsarXiv:2408.08022
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Abstract

We prove a sharp quartic curvature pinching for the mean curvature flow in , , which generalises Pu's work on the convergence of submanifolds in to a round point. Using a blow up argument, we prove a codimension and a cylindrical estimate, where in regions of high curvature, the submanifold becomes approximately codimension one, quantitatively, and is weakly convex and moves by translation or is a self shrinker. With a decay estimate, the rescaling converges smoothly to a totally geodesic limit in infinite time, without using Stampacchia iteration or integral analysis.

Mean curvature flowSelf-shrinkers & translatorsSingularitiesCurvature pinchingLong-time existence & convergence

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 Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds (Artemis A. Vogiatzi and Huy T. Nguyen, 2023)2Vogiatzi 2023 Mean Curvature Flow of High Codimension in Complex Projective Space (Artemis A. Vogiatzi, 2023)1Vogiatzi 2023 A gap Theorem on closed self-shrinkers of mean curvature flow (Yuhang Zhao, 2025)3Zhao 2025 Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows (Mat Langford and James McCoy, 2025)6Langford 2025 Classification of quadratically pinched self-shrinkers in higher codimension (Debora Impera, Michele Rimoldi and Francesco Ruatta, 2026)4Impera 2026 High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension (Ben Andrews and Qiyu Zhou, 2026)7Andrews 2026 The rigidity theorems for complete self-expander of mean curvature flow (Zhi Li and Guoxin Wei, 2026)8Li 2026 The planarity estimate in dimensions five and six (Stephen Lynch, 2026)5Lynch 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean Curvature Flow of High Codimension in Complex Projective Space

    Artemis A. VogiatziWider flows PDF

    Topics Curvature pinching, Long-time existence & convergence, Singularities · Same author Artemis A. Vogiatzi · Shared terms “high curvature”, “approximately codimension”, “becomes approximately”

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

    Artemis A. Vogiatzi and Huy T. NguyenWider flows PDF

    Topics Curvature pinching, Singularities · Same author Artemis A. Vogiatzi · Shared terms “high curvature”, “approximately codimension”, “becomes approximately”

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

    Yuhang ZhaoWider flows PDF

    Topics Curvature pinching, Self-shrinkers & translators

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

    Debora Impera, Michele Rimoldi and Francesco RuattaWider flows PDF

    Topics Curvature pinching, Self-shrinkers & translators

  5. The planarity estimate in dimensions five and six

    Stephen LynchWider flows PDF

    Topics Curvature pinching, Singularities · Shared terms “high curvature”

  6. Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows

    Mat Langford and James McCoyWider flows PDF

    Topics Curvature pinching · Shared terms “cylindrical estimate”, “Stampacchia iteration”

  7. High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension

    Ben Andrews and Qiyu ZhouWider flows PDF

    Topics Curvature pinching, Singularities · Shared terms “submanifolds”

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

    Zhi Li and Guoxin WeiWider flows PDF

    Topics Curvature pinching, Self-shrinkers & translators

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, 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)

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.