Paper · Apr 17, 2020

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

Keaton Naff

math.DGWider flowsarXiv:2004.08060v3
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Abstract

In dimensions , we prove a canonical neighborhood theorem for the mean curvature flow of compact -dimensional submanifolds in satisfying a pinching condition for

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Submitted
Apr 17, 2020
Last revised
Jul 13, 2022 (v3)
Subjects
math.DG

Comments: final version, to appear in IMRN

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 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 A planarity estimate for pinched solutions of mean curvature flow (Keaton Naff, 2019)3Naff 2019 Singularity models of pinched solutions of mean curvature flow in higher codimension (Keaton Naff, 2019)2Naff 2019 Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds (Artemis A. Vogiatzi and Huy T. Nguyen, 2023)6Vogiatzi 2023 On κ-solutions and canonical neighborhoods in 4d Ricci flow (Robert Haslhofer, 2023)7Haslhofer 2023 Classification of quadratically pinched self-shrinkers in higher codimension (Debora Impera, Michele Rimoldi and Francesco Ruatta, 2026)4Impera 2026 A mean curvature flow approach to Hamilton's pinching theorem (Liang Cheng and Zhenyu Lu, 2026)8Cheng 2026 The rigidity theorems for complete self-expander of mean curvature flow (Zhi Li and Guoxin Wei, 2026)5Li 2026 Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow (Tsz-Kiu Aaron Chow, 2026)1Chow 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow

    Tsz-Kiu Aaron Chow

    Topics Curvature pinching · Shared terms “higher codimension”, “neighborhood theorem”, “canonical neighborhood”

  2. Singularity models of pinched solutions of mean curvature flow in higher codimension

    Keaton NaffWider flows

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

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

    Keaton NaffWider flows

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

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

    Debora Impera, Michele Rimoldi and Francesco RuattaWider flows

    Topics Curvature pinching · Shared terms “higher codimension”, “pinching condition”

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

    Zhi Li and Guoxin WeiWider flows

    Topics Curvature pinching · Shared terms “higher codimension”, “pinching condition”

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

    Artemis A. Vogiatzi and Huy T. NguyenWider flows

    Topics Curvature pinching · Shared terms “pinching condition”

  7. On -solutions and canonical neighborhoods in 4d Ricci flow

    Robert Haslhofer

    Topics Mean curvature flow · Shared terms “canonical neighborhood”, “neighborhood theorem”

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

    Liang Cheng and Zhenyu LuWider flows

    Topics Curvature 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 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)

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.