Paper · Sep 8, 2023

Collapsing of Mean Curvature Flow of Hypersurfaces to Complex Submanifolds

Farnaz Ghanbari, Samreena

math.DGWider flowsarXiv:2309.04176
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As on arXiv

Abstract

In this paper, we produce explicit examples of mean curvature flow of (2m-1)-dimensional submanifolds which converge to (2m-2)-dimensional submanifolds at a finite time. These examples are a special class of hyperspheres in with a -invariant Kähler metrics. We first discuss the mean curvature flow problem and then investigate the type of singularities for them.

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Submitted
Sep 8, 2023
Subjects
math.DG
Mean curvature flowSingularities(Non)collapsingKähler geometry

Its neighbourhood

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I

    Tongxin Xu and Zhenlei Zhang

    Topics (Non)collapsing, Kähler geometry, Singularities · Shared terms “finite time”

  2. Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces

    Tongxin Xu and Zhenlei Zhang

    Topics (Non)collapsing, Kähler geometry, Singularities · Shared terms “finite time”

  3. Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I

    Charles Cifarelli, Ronan Conlon, Max Hallgren et al.

    Topics (Non)collapsing, Kähler geometry, Singularities · Shared terms “finite time”

  4. Mean curvature flow through singularities

    Robert HaslhoferWider flows

    Topics (Non)collapsing, Singularities

  5. Non-collapsed finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I

    Ronan J. Conlon, Max Hallgren and Zilu Ma

    Topics (Non)collapsing, Kähler geometry, Singularities · Shared terms “finite time”

  6. High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension

    Ben Andrews and Qiyu ZhouWider flows

    Topics (Non)collapsing, Singularities · Shared terms “submanifolds”, “finite time”

  7. On the tangent flow to the collapsing Kähler-Ricci flow on Hirzebruch surfaces

    Jiangtao Li

    Topics (Non)collapsing, Kähler geometry, Singularities · Shared terms “finite time”

  8. Finite Time Singularities of the Kähler Ricci Flow on Compact Kähler Surfaces are of Type I

    Yeyun Xu and Linfeng Zhou

    Topics (Non)collapsing, Kähler geometry, Singularities · Shared terms “finite time”

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.