Paper · Jun 19, 2019

A planarity estimate for pinched solutions of mean curvature flow

Keaton Naff

math.DGWider flowsarXiv:1906.08184v2
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As on arXiv

Abstract

We show that the blow-ups of compact solutions to the mean curvature flow in initially satisfying the pinching condition for a suitable constant must be codimension one.

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Submitted
Jun 19, 2019
Last revised
Mar 23, 2021 (v2)
Subjects
math.DG

Comments: title changed, final version, accepted to Duke Math J

Mean curvature flowSingularitiesCurvature 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 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 Mean curvature flow for pinched submanifolds in rank one symmetric spaces (Naoyuki Koike, Yoshiyuki Mizumura and Nana Uenoyama, 2017)8Koike 2017 Singularity models of pinched solutions of mean curvature flow in higher codimension (Keaton Naff, 2019)1Naff 2019 A canonical neighborhood theorem for the mean curvature flow in higher codimension (Keaton Naff, 2020)4Naff 2020 Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds (Artemis A. Vogiatzi and Huy T. Nguyen, 2023)7Vogiatzi 2023 Planarity and convexity for pinched ancient solutions of mean curvature flow (Tang-Kai Lee, Keaton Naff and Jingze Zhu, 2025)6Lee 2025 The planarity estimate in dimensions five and six (Stephen Lynch, 2026)2Lynch 2026 Local Convexity and Planarity Estimates for Mean Curvature Flow (Liang Cheng, 2026)3Cheng 2026 Planarity estimates and codimension reduction for ancient flows at the Andrews-Baker threshold (Francesco Ruatta, 2026)5Ruatta 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singularity models of pinched solutions of mean curvature flow in higher codimension

    Keaton NaffWider flows

    Topics Curvature pinching, Singularities · Same author Keaton Naff · Shared terms “pinched solutions”, “pinching condition”

  2. The planarity estimate in dimensions five and six

    Stephen LynchWider flows

    Topics Curvature pinching, Singularities · Shared terms “planarity estimate”

  3. Local Convexity and Planarity Estimates for Mean Curvature Flow

    Liang ChengWider flows

    Topics Curvature pinching · Shared terms “planarity estimate”, “pinching condition”

  4. A canonical neighborhood theorem for the mean curvature flow in higher codimension

    Keaton NaffWider flows

    Topics Curvature pinching · Same author Keaton Naff · Shared terms “pinching condition”

  5. Planarity estimates and codimension reduction for ancient flows at the Andrews-Baker threshold

    Francesco RuattaWider flows

    Topics Curvature pinching · Shared terms “planarity estimate”

  6. Planarity and convexity for pinched ancient solutions of mean curvature flow

    Tang-Kai Lee, Keaton Naff and Jingze Zhu

    Topics Mean curvature flow · Same author Keaton Naff · Shared terms “planarity”

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

    Artemis A. Vogiatzi and Huy T. NguyenWider flows

    Topics Curvature pinching, Singularities · Shared terms “pinching condition”

  8. 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 “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) 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)

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.