Paper · Mar 31, 2016

On the rigidity of mean convex self-shrinkers

Qiang Guang, Jonathan J. Zhu

math.DGWider flowsarXiv:1603.09435
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Abstract

Self-shrinkers model singularities of the mean curvature flow; they are defined as the special solutions that contract homothetically under the flow. Colding-Ilmanen-Minicozzi showed that cylindrical self-shrinkers are rigid in a strong sense - that is, any self-shrinker that is mean convex with uniformly bounded curvature on a large, but compact, set must be a round cylinder. Using this result, Colding and Minicozzi were able to establish uniqueness of blowups at cylindrical singularities, and provide a detailed description of the singular set of generic mean curvature flows. In this paper, we show that the bounded curvature assumption is unnecessary for the rigidity of the cylinder if either n is at most 6, or if the mean curvature is bounded below by a positive constant. These results follow from curvature estimates that we prove for strictly mean convex self-shrinkers. We also obtain a rigidity theorem in all dimensions for graphical self-shrinkers, and curvature estimates for translators of the mean curvature flow.

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Submitted
Mar 31, 2016
Subjects
math.DG
Journal
Int. Math. Res. Not. IMRN 2018, no. 20, 6406-6425
DOI
10.1093/imrn/rnx078

Comments: 15 pages, comments welcome!

Mean curvature flowSelf-shrinkers & translatorsSingularitiesStability & uniqueness

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)1Zhu 2016 Volume growth, entropy and stability for translating solitons (Qiang Guang, 2016)6Guang 2016 Uniqueness of asymptotically conical tangent flows (Otis Chodosh and Felix Schulze, 2019)7Chodosh 2019 Łojasiewicz inequalities, uniqueness and rigidity for cylindrical self-shrinkers (Jonathan J. Zhu, 2020)2Zhu 2020 Rigidity and Łojasiewicz inequalities for Clifford self-shrinkers (Ao Sun and Jonathan J. Zhu, 2020)4Sun 2020 Mean curvature flow with generic initial data II (Otis Chodosh, Kyeongsu Choi and Felix Schulze, 2023)8Chodosh 2023 Uniqueness of blowups for forced mean curvature flow (Sven Hirsch and Jonathan J. Zhu, 2023)3Hirsch 2023 A stable self-shrinking Möbius bundle in ℝ⁴ (Kahnrad Braxton, Tang-Kai Lee and Jonathan J. Zhu, 2026)5Braxton 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 entropy of closed hypersurfaces and singular self-shrinkers

    Jonathan J. ZhuWider flows

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

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

    Jonathan J. Zhu

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

  3. Uniqueness of blowups for forced mean curvature flow

    Sven Hirsch and Jonathan J. ZhuWider flows

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

  4. Rigidity and Łojasiewicz inequalities for Clifford self-shrinkers

    Ao Sun and Jonathan J. Zhu

    Topics Self-shrinkers & translators, Stability & uniqueness · Same author Jonathan J. Zhu

  5. A stable self-shrinking Möbius bundle in

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

    Topics Self-shrinkers & translators, Singularities, Stability & uniqueness · Same author Jonathan J. Zhu · Shared terms “shrinker models”

  6. Volume growth, entropy and stability for translating solitons

    Qiang GuangWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Same author Qiang Guang · Shared terms “curvature estimates”

  7. Uniqueness of asymptotically conical tangent flows

    Otis Chodosh and Felix SchulzeWider flows

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

  8. Mean curvature flow with generic initial data II

    Otis Chodosh, Kyeongsu Choi and Felix SchulzeWider flows

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

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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

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    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.