Paper · Oct 18, 2018

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

Ao Sun

math.DGWider flowsarXiv:1810.08114v3
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Abstract

In this paper we prove that the generic singularities of mean curvature flow of closed embedded surfaces in modeled by closed self-shrinkers with multiplicity has multiplicity one. Together with the previous result by Colding-Minicozzi in [CM12], we conclude that the only generic singularity of mean curvature flow of closed embedded surfaces in modeled by closed self-shrinkers is a multiplicity one sphere. We also construct particular perturbations of the flow to avoid those singularities with multiplicity higher than one. Our result partially addresses the well-known multiplicity one conjecture by Ilmanen.

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Submitted
Oct 18, 2018
Last revised
Jul 23, 2021 (v3)
Subjects
math.DG

Comments: 26 pages. Correct some typos and revise the paper according to the suggestions by the referees. Accepted by Journal of Differential Geometry

Mean curvature flowSelf-shrinkers & translatorsSingularities

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 On the entropy of closed hypersurfaces and singular self-shrinkers (Jonathan J. Zhu, 2016)5Zhu 2016 Singularities of Mean Curvature Flow of Surfaces with Additional Forces (Ao Sun, 2018)2Sun 2018 Uniqueness of asymptotically conical tangent flows (Otis Chodosh and Felix Schulze, 2019)8Chodosh 2019 Bifurcation of perturbations of non-generic closed self-shrinkers (Zhengjiang Lin and Ao Sun, 2020)1Lin 2020 Closed embedded self-shrinkers of mean curvature flow (Oskar Riedler, 2022)7Riedler 2022 On mean curvature flow translators with prescribed ends (Ao Sun and Zhihan Wang, 2023)6Sun 2023 Mean curvature flow with generic initial data II (Otis Chodosh, Kyeongsu Choi and Felix Schulze, 2023)4Chodosh 2023 Genus one singularities in mean curvature flow (Adrian Chun-Pong Chu and Ao Sun, 2023)3Chu 2023
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Bifurcation of perturbations of non-generic closed self-shrinkers

    Zhengjiang Lin and Ao SunWider flows

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

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

    Ao SunWider flows

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

  3. 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”

  4. Mean curvature flow with generic initial data II

    Otis Chodosh, Kyeongsu Choi and Felix SchulzeWider flows

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

  5. On the entropy of closed hypersurfaces and singular self-shrinkers

    Jonathan J. ZhuWider flows

    Topics Self-shrinkers & translators, Singularities · Shared terms “Colding Minicozzi”, “Ilmanen”

  6. On mean curvature flow translators with prescribed ends

    Ao Sun and Zhihan WangWider flows

    Topics Self-shrinkers & translators · Same author Ao Sun · Shared terms “closed embedded”, “Ilmanen”

  7. Closed embedded self-shrinkers of mean curvature flow

    Oskar RiedlerWider flows

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

  8. Uniqueness of asymptotically conical tangent flows

    Otis Chodosh and Felix SchulzeWider flows

    Topics Self-shrinkers & translators, Singularities · Shared terms “multiplicity”, “embedded surfaces”, “Colding Minicozzi”

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)

In the guidebook

  1. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.