Paper · Sep 27, 2017

Pinched Ancient Solutions to the High Codimension Mean Curvature Flow

Stephen Lynch, Huy The Nguyen

math.DGWider flowsarXiv:1709.09697
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Abstract

We study solutions of high codimension mean curvature flow defined for all negative times, usually referred to as ancient solutions. We show that any compact ancient solution whose second fundamental form satisfies a certain natural pinching condition must be a family of shrinking spheres. Andrews and Baker have shown that initial submanifolds satisfying this pinching condition, which generalises the notion of convexity, converge to round points under the flow. As an application, we use our result to simplify their proof.

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Submitted
Sep 27, 2017
Subjects
math.DG
Mean curvature flowAncient solutionsCurvature 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 Ancient solutions of geometric flows with curvature pinching (Susanna Risa and Carlo Sinestrari, 2017)6Risa 2017 Cylindrical Estimates for High Codimension Mean Curvature Flow (Huy The Nguyen, 2018)8Nguyen 2018 High Codimension Mean Curvature Flow with Surgery (Stephen Lynch and Huy The Nguyen, 2020)1Lynch 2020 Evolving Pinched Submanifolds of the Sphere by Mean Curvature Flow (Charles Baker and Huy The Nguyen, 2020)4Baker 2020 Convexity Estimates for High Codimension Mean Curvature Flow (Stephen Lynch and Huy The Nguyen, 2020)3Lynch 2020 Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery (Mat Langford, Stephen Lynch and Huy The Nguyen, 2021)5Langford 2021 Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds (Artemis A. Vogiatzi and Huy T. Nguyen, 2023)2Vogiatzi 2023 The planarity estimate in dimensions five and six (Stephen Lynch, 2026)7Lynch 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. High Codimension Mean Curvature Flow with Surgery

    Stephen Lynch and Huy The NguyenWider flows

    Topics Curvature pinching · Same authors Stephen Lynch, Huy The Nguyen · Shared terms “high codimension”, “codimension mean”, “submanifolds satisfying”

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

    Artemis A. Vogiatzi and Huy T. NguyenWider flows

    Topics Curvature pinching · Same author Huy The Nguyen · Shared terms “high codimension”, “codimension mean”, “pinching condition”

  3. Convexity Estimates for High Codimension Mean Curvature Flow

    Stephen Lynch and Huy The NguyenWider flows

    Topics Mean curvature flow · Same authors Stephen Lynch, Huy The Nguyen · Shared terms “high codimension”, “codimension mean”, “fundamental form”

  4. Evolving Pinched Submanifolds of the Sphere by Mean Curvature Flow

    Charles Baker and Huy The NguyenWider flows

    Topics Curvature pinching · Same author Huy The Nguyen · Shared terms “codimension mean”, “high codimension”, “pinching condition”

  5. Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery

    Mat Langford, Stephen Lynch and Huy The NguyenWider flows

    Topics Curvature pinching · Same authors Stephen Lynch, Huy The Nguyen · Shared terms “pinching condition”

  6. Ancient solutions of geometric flows with curvature pinching

    Susanna Risa and Carlo SinestrariWider flows

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

  7. The planarity estimate in dimensions five and six

    Stephen LynchWider flows

    Topics Curvature pinching · Same author Stephen Lynch · Shared terms “high codimension”, “Baker”, “Andrews”

  8. Cylindrical Estimates for High Codimension Mean Curvature Flow

    Huy The NguyenWider flows

    Topics Mean curvature flow · Same author Huy The Nguyen · Shared terms “high codimension”, “codimension mean”

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) 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.