Paper · Apr 16, 2020

Bifurcation of perturbations of non-generic closed self-shrinkers

Zhengjiang Lin, Ao Sun

math.DGWider flowsarXiv:2004.07787v3
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Abstract

We discover a bifurcation of the perturbations of non-generic closed self-shrinkers. If the generic perturbation is outward, then the next mean curvature flow singularity is cylindrical and collapsing from outside; if the generic perturbation is inward, then the next mean curvature flow singularity is cylindrical and collapsing from inside.

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Submitted
Apr 16, 2020
Last revised
Dec 8, 2020 (v3)
Subjects
math.DG

Comments: 21 pages, 1 figures; We rewrite our proof of theorem 3.1 about the finite time singularity in this version. To appear in Journal of Topology and Analysis

Mean curvature flowSelf-shrinkers & translatorsSingularities(Non)collapsing

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. Local Entropy and Generic Multiplicity One Singularities of Mean Curvature Flow of Surfaces

    Ao SunWider flows

    Topics Self-shrinkers & translators, Singularities · Same author Ao Sun · Shared terms “closed self”, “perturbation”

  2. Genus one singularities in mean curvature flow

    Adrian Chun-Pong Chu and Ao SunWider flows

    Topics Self-shrinkers & translators, Singularities · Same author Ao Sun · Shared terms “perturbation”

  3. Mean curvature flow with generic initial data II

    Otis Chodosh, Kyeongsu Choi and Felix SchulzeWider flows

    Topics Self-shrinkers & translators, Singularities · Shared terms “generic closed”

  4. Singularities of Mean Curvature Flow of Surfaces with Additional Forces

    Ao SunWider flows

    Topics Self-shrinkers & translators, Singularities · Same author Ao Sun

  5. A gap Theorem on closed self-shrinkers of mean curvature flow

    Yuhang ZhaoWider flows

    Topics Self-shrinkers & translators · Shared terms “closed self”

  6. A survey of closed self-shrinkers with symmetry

    Gregory Drugan, Hojoo Lee and Xuan Hien NguyenWider flows

    Topics Self-shrinkers & translators · Shared terms “closed self”

  7. On mean curvature flow translators with prescribed ends

    Ao Sun and Zhihan WangWider flows

    Topics Self-shrinkers & translators · Same author Ao Sun

  8. Rigidity and Łojasiewicz inequalities for Clifford self-shrinkers

    Ao Sun and Jonathan J. Zhu

    Topics Self-shrinkers & translators · Same author Ao Sun

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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) 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) (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)

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.