Paper · Oct 9, 2019

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

Keaton Naff

math.DGWider flowsarXiv:1910.03968v2
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Abstract

We consider ancient solutions to the mean curvature flow in () that are weakly convex, uniformly two-convex, and satisfy derivative estimates . We show that such solutions are noncollapsed. As an application, in arbitrary codimension, we consider compact -dimensional () solutions to the mean curvature flow in that satisfy the pinching condition and , . We conclude that any blow-up model at the first singular time must be a codimension one shrinking sphere, shrinking cylinder, or translating bowl soliton.

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Submitted
Oct 9, 2019
Last revised
Apr 17, 2020 (v2)
Subjects
math.DG

Comments: We have added a structure theorem for the case when the ancient solution is compact

Mean curvature flowAncient solutionsSelf-shrinkers & translatorsSingularities(Non)collapsingCurvature 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 Pinched Ancient Solutions to the High Codimension Mean Curvature Flow (Stephen Lynch and Huy The Nguyen, 2017)5Lynch 2017 A planarity estimate for pinched solutions of mean curvature flow (Keaton Naff, 2019)1Naff 2019 A canonical neighborhood theorem for the mean curvature flow in higher codimension (Keaton Naff, 2020)2Naff 2020 Uniqueness of compact ancient solutions to the higher dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al., 2021)6Brendle 2021 Collapsing and noncollapsing in convex ancient mean curvature flow (Theodora Bourni, Mat Langford and Stephen Lynch, 2021)4Bourni 2021 Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds (Artemis A. Vogiatzi and Huy T. Nguyen, 2023)8Vogiatzi 2023 Planarity and convexity for pinched ancient solutions of mean curvature flow (Tang-Kai Lee, Keaton Naff and Jingze Zhu, 2025)7Lee 2025 Classification of quadratically pinched self-shrinkers in higher codimension (Debora Impera, Michele Rimoldi and Francesco Ruatta, 2026)3Impera 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A planarity estimate for pinched solutions of mean curvature flow

    Keaton NaffWider flows

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

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

    Keaton NaffWider flows

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

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

    Debora Impera, Michele Rimoldi and Francesco RuattaWider flows

    Topics Curvature pinching, Ancient solutions, Self-shrinkers & translators · Shared terms “higher codimension”, “pinching condition”, “shrinking cylinder”

  4. Collapsing and noncollapsing in convex ancient mean curvature flow

    Theodora Bourni, Mat Langford and Stephen LynchWider flows

    Topics (Non)collapsing, Ancient solutions, Singularities · Shared terms “pinched solutions”, “singularity models”, “higher codimension”

  5. Pinched Ancient Solutions to the High Codimension Mean Curvature Flow

    Stephen Lynch and Huy The NguyenWider flows

    Topics Curvature pinching, Ancient solutions · Shared terms “pinching condition”, “shrinking sphere”

  6. Uniqueness of compact ancient solutions to the higher dimensional Ricci flow

    Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al.

    Topics (Non)collapsing, Ancient solutions, Singularities · Same author Keaton Naff · Shared terms “noncollapsed”

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

    Tang-Kai Lee, Keaton Naff and Jingze Zhu

    Topics Ancient solutions · Same author Keaton Naff · Shared terms “higher codimension”

  8. 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 “singularity models”, “weakly convex”, “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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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 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)

In the guidebook

  1. 12B.4
    Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery

    Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.

  2. 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).

  3. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.