Paper · Mar 1, 2017

Mean curvature flow for pinched submanifolds in rank one symmetric spaces

Naoyuki Koike, Yoshiyuki Mizumura, Nana Uenoyama

math.DGWider flowsarXiv:1703.00202v5
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Abstract

G. Pipoli and C. Sinestrari considered the mean curvature flow starting from a closed submanifold in the complex projective space. They proved that if the submanifold is of small codimension and satisfies a suitable pinching condition for the second fundamental form, then the flow has two possible behaviors: either the submanifold collapses to a round point in finite time, or it converges smoothly to a totally geodesic submanifold in infinite time. In this paper, we prove the similar results for the mean curvature flow starting from pinched closed submanifolds in (general) rank one symmetric spaces of compact type. Also, we prove that closed submanifolds in (general) rank one symmetric spaces of non-compact type collapse to a round point along the mean curvature flow under certain strict pinching condition for the norm of the second fundamental form.

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Submitted
Mar 1, 2017
Last revised
Dec 14, 2020 (v5)
Subjects
math.DG

Comments: In this paper, we found some important misses. Hence we decided to withdrawal this manuscript

Mean curvature flowSingularitiesCurvature pinchingBoundaries & non-compactLong-time existence & convergence

Its neighbourhood

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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 Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact type (Naoyuki Koike, 2016)1Koike 2016 Mean curvature flow of arbitrary codimension in complex projective spaces (Li Lei and Hongwei Xu, 2016)2Lei 2016 Pinched Ancient Solutions to the High Codimension Mean Curvature Flow (Stephen Lynch and Huy The Nguyen, 2017)8Lynch 2017 A planarity estimate for pinched solutions of mean curvature flow (Keaton Naff, 2019)7Naff 2019 Evolving Pinched Submanifolds of the Sphere by Mean Curvature Flow (Charles Baker and Huy The Nguyen, 2020)3Baker 2020 Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery (Mat Langford, Stephen Lynch and Huy The Nguyen, 2021)5Langford 2021 High Codimension Mean Curvature Flow in the Complex Projective Space (Artemis A. Vogiatzi, 2023)4Vogiatzi 2023 Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere (Artemis A. Vogiatzi, 2024)6Vogiatzi 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact type

    Naoyuki KoikeWider flows

    Topics Boundaries & non-compact, Singularities · Same author Naoyuki Koike · Shared terms “symmetric spaces”, “compact type”, “infinite time”

  2. Mean curvature flow of arbitrary codimension in complex projective spaces

    Li Lei and Hongwei XuWider flows

    Topics Curvature pinching, Singularities · Shared terms “complex projective”, “projective space”, “geodesic submanifold”

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

    Charles Baker and Huy The NguyenWider flows

    Topics Curvature pinching · Shared terms “pinched submanifold”, “pinching condition”, “round point”

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

    Artemis A. VogiatziWider flows

    Topics Curvature pinching, Long-time existence & convergence, Singularities · Shared terms “complex projective”, “projective space”, “infinite time”

  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 · Shared terms “pinched submanifold”, “pinching condition”, “totally geodesic”

  6. Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere

    Artemis A. VogiatziWider flows

    Topics Curvature pinching, Long-time existence & convergence, Singularities · Shared terms “round point”, “infinite time”, “totally geodesic”

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

    Keaton NaffWider flows

    Topics Curvature pinching, Singularities · Shared terms “pinching condition”

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

    Stephen Lynch and Huy The NguyenWider flows

    Topics Curvature pinching · Shared terms “pinching condition”, “round point”, “fundamental form”

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) 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) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 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).

  2. 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.

  3. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.