Paper · Jun 9, 2020

Convexity Estimates for High Codimension Mean Curvature Flow

Stephen Lynch, Huy The Nguyen

math.DGWider flowsarXiv:2006.05227
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Abstract

We consider the evolution by mean curvature of smooth -dimensional submanifolds in which are compact and quadratically pinched. We will be primarily interested in flows of high codimension, the case . We prove that our submanifold is asymptotically convex, that is the first eigenvalue of the second fundamental form in the principal mean curvature direction blows up at a strictly slower rate than the mean curvature vector. We use this convexity estimate to show that at a singular time of the flow, there exists a rescaling that converges to a smooth codimension-one limiting flow which is convex and moves by translation.

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Submitted
Jun 9, 2020
Subjects
math.DG
Mean curvature flowEigenvalues & spectrum

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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)2Lynch 2017 Cylindrical Estimates for High Codimension Mean Curvature Flow (Huy The Nguyen, 2018)3Nguyen 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)7Baker 2020 Convexity estimates for hypersurfaces moving by concave curvature functions (Stephen Lynch, 2020)6Lynch 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)4Vogiatzi 2023 High Codimension Mean Curvature Flow in the Complex Projective Space (Artemis A. Vogiatzi, 2023)8Vogiatzi 2023
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 Mean curvature flow · Same authors Stephen Lynch, Huy The Nguyen · Shared terms “high codimension”, “codimension mean”, “quadratically”

  2. Pinched Ancient Solutions to the 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”

  3. 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”

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

    Artemis A. Vogiatzi and Huy T. NguyenWider flows

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

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

    Mat Langford, Stephen Lynch and Huy The NguyenWider flows

    Topics Mean curvature flow · Same authors Stephen Lynch, Huy The Nguyen · Shared terms “quadratically pinched”, “dimensional submanifolds”

  6. Convexity estimates for hypersurfaces moving by concave curvature functions

    Stephen LynchWider flows

    Topics Mean curvature flow · Same author Stephen Lynch · Shared terms “convexity estimate”, “principal”

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

    Charles Baker and Huy The NguyenWider flows

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

  8. High Codimension Mean Curvature Flow in the Complex Projective Space

    Artemis A. VogiatziWider flows

    Topics Mean curvature flow · Shared terms “high codimension”, “codimension mean”, “limiting flow”

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) Eigenvalues & spectrum Perelman's λ-functional is the lowest eigenvalue of −4Δ+ R, and it never decreases along Ricci flow. Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

  1. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.