Paper · Jul 26, 2016

On the entropy of closed hypersurfaces and singular self-shrinkers

Jonathan J. Zhu

math.DGWider flowsarXiv:1607.07760
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Abstract

Self-shrinkers are the special solutions of mean curvature flow in that evolve by shrinking homothetically; they serve as singularity models for the flow. The entropy of a hypersurface introduced by Colding-Minicozzi is a Lyapunov functional for the mean curvature flow, and is fundamental to their theory of generic mean curvature flow. In this paper we prove that a conjecture of Colding-Ilmanen-Minicozzi-White, namely that any closed hypersurface in has entropy at least that of the round sphere, holds in any dimension . This result had previously been established for the cases by Bernstein-Wang using a carefully constructed weak flow. The main technical result of this paper is an extension of Colding-Minicozzi's classification of entropy-stable self-shrinkers to the singular setting. In particular, we show that any entropy-stable self-shrinker whose singular set satisfies Wickramasekera's -structural hypothesis must be a round cylinder .

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Submitted
Jul 26, 2016
Subjects
math.DG

Comments: 35 pages, comments welcome!

Mean curvature flowSelf-shrinkers & translatorsSingularitiesWeak & singular flows

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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 On the rigidity of mean convex self-shrinkers (Qiang Guang and Jonathan J. Zhu, 2016)1Guang 2016 Local Entropy and Generic Multiplicity One Singularities of Mean Curvature Flow of Surfaces (Ao Sun, 2018)5Sun 2018 Łojasiewicz inequalities, uniqueness and rigidity for cylindrical self-shrinkers (Jonathan J. Zhu, 2020)3Zhu 2020 Rigidity and Łojasiewicz inequalities for Clifford self-shrinkers (Ao Sun and Jonathan J. Zhu, 2020)7Sun 2020 Uniqueness of blowups for forced mean curvature flow (Sven Hirsch and Jonathan J. Zhu, 2023)2Hirsch 2023 Closed mean curvature flows with asymptotically conical singularities (Tang-Kai Lee and Xinrui Zhao, 2024)8Lee 2024 Li-Yau Conformal Volume and Colding-Minicozzi Entropy of Self-Shrinkers (Jacob Bernstein, 2024)6Bernstein 2024 A stable self-shrinking Möbius bundle in ℝ⁴ (Kahnrad Braxton, Tang-Kai Lee and Jonathan J. Zhu, 2026)4Braxton 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the rigidity of mean convex self-shrinkers

    Qiang Guang and Jonathan J. ZhuWider flows

    Topics Self-shrinkers & translators, Singularities · Same author Jonathan J. Zhu · Shared terms “Colding Ilmanen”, “Ilmanen Minicozzi”, “special solutions”

  2. Uniqueness of blowups for forced mean curvature flow

    Sven Hirsch and Jonathan J. ZhuWider flows

    Topics Self-shrinkers & translators, Singularities · Same author Jonathan J. Zhu · Shared terms “Colding Minicozzi”

  3. Łojasiewicz inequalities, uniqueness and rigidity for cylindrical self-shrinkers

    Jonathan J. Zhu

    Topics Self-shrinkers & translators, Singularities · Same author Jonathan J. Zhu · Shared terms “round cylinder”, “Colding Minicozzi”

  4. A stable self-shrinking Möbius bundle in

    Kahnrad Braxton, Tang-Kai Lee and Jonathan J. Zhu

    Topics Self-shrinkers & translators, Singularities · Same author Jonathan J. Zhu · Shared terms “stable self”

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

    Ao SunWider flows

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

  6. Li-Yau Conformal Volume and Colding-Minicozzi Entropy of Self-Shrinkers

    Jacob BernsteinWider flows

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

  7. Rigidity and Łojasiewicz inequalities for Clifford self-shrinkers

    Ao Sun and Jonathan J. Zhu

    Topics Self-shrinkers & translators · Same author Jonathan J. Zhu · Shared terms “previously”

  8. Closed mean curvature flows with asymptotically conical singularities

    Tang-Kai Lee and Xinrui Zhao

    Topics Weak & singular flows, Self-shrinkers & translators, Singularities · Shared terms “closed hypersurface”, “Ilmanen”

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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 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.

  3. 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.