Paper · Apr 26, 2020

Evolving Pinched Submanifolds of the Sphere by Mean Curvature Flow

Charles Baker, Huy The Nguyen

math.DGWider flowsarXiv:2004.12259
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Abstract

In this paper, we prove convergence of the high codimension mean curvature flow in the sphere to either a round point or a totally geodesic sphere assuming a pinching condition between the norm squared of the second fundamental form and the norm squared of the mean curvature and the background curvature of the sphere. We show that this pinching is sharp for dimension but is not sharp for dimension . For dimension and codimension , we consider an alternative pinching condition which includes the normal curvature of the normal bundle. Finally, we sharpen the Chern-do Carmo-Kobayashi curvature condition for surfaces in the four sphere - this curvature condition is sharp for minimal surfaces and we conjecture it to be sharp for curvature flows in the sphere.

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Submitted
Apr 26, 2020
Subjects
math.DG, math.AP
Mean curvature flowOther curvature flowsCurvature pinchingMinimal & CMC surfaces

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 Surfaces of Co-dimension Two Pinched by Normal Curvature (Charles Baker and Huy The Nguyen, 2016)2Baker 2016 Mean curvature flow for pinched submanifolds in rank one symmetric spaces (Naoyuki Koike, Yoshiyuki Mizumura and Nana Uenoyama, 2017)8Koike 2017 Pinched Ancient Solutions to the High Codimension Mean Curvature Flow (Stephen Lynch and Huy The Nguyen, 2017)3Lynch 2017 High Codimension Mean Curvature Flow with Surgery (Stephen Lynch and Huy The Nguyen, 2020)7Lynch 2020 Quadratically pinched hypersurfaces of the sphere via mean curvature flow with surgery (Mat Langford and Huy The Nguyen, 2020)4Langford 2020 Sharp pinching estimates for mean curvature flow in the sphere (Mat Langford and Huy The Nguyen, 2020)6Langford 2020 Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery (Mat Langford, Stephen Lynch and Huy The Nguyen, 2021)1Langford 2021 Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds (Artemis A. Vogiatzi and Huy T. Nguyen, 2023)5Vogiatzi 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery

    Mat Langford, Stephen Lynch and Huy The NguyenWider flows

    Topics Curvature pinching, Minimal & CMC surfaces · Same author Huy The Nguyen · Shared terms “pinched submanifold”, “pinching condition”, “totally geodesic”

  2. Surfaces of Co-dimension Two Pinched by Normal Curvature

    Charles Baker and Huy The NguyenWider flows

    Topics Mean curvature flow · Same authors Charles Baker, Huy The Nguyen · Shared terms “normal curvature”, “curvature condition”, “round point”

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

    Stephen Lynch and Huy The NguyenWider flows

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

  4. Quadratically pinched hypersurfaces of the sphere via mean curvature flow with surgery

    Mat Langford and Huy The NguyenWider flows

    Topics Curvature pinching, Minimal & CMC surfaces · Same author Huy The Nguyen · Shared terms “pinching condition”, “totally geodesic”

  5. 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”

  6. Sharp pinching estimates for mean curvature flow in the sphere

    Mat Langford and Huy The NguyenWider flows

    Topics Curvature pinching, Minimal & CMC surfaces · Same author Huy The Nguyen · Shared terms “fundamental form”

  7. High Codimension Mean Curvature Flow with Surgery

    Stephen Lynch and Huy The NguyenWider flows

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

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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

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.