Paper · Nov 3, 2020

Łojasiewicz inequalities, uniqueness and rigidity for cylindrical self-shrinkers

Jonathan J. Zhu

math.DGarXiv:2011.01633v2
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Abstract

We establish Łojasiewicz inequalities for a class of cylindrical self-shrinkers for the mean curvature flow, which includes round cylinders and cylinders over Abresch-Langer curves, in any codimension. We deduce the uniqueness of blowups at singularities modelled on this class of cylinders, and that any such cylinder is isolated in the space of self-shrinkers. The Abresch-Langer case answers a conjecture of Colding-Minicozzi. Our proof uses direct perturbative analysis of the shrinker mean curvature, so it is new even for round cylinders.

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Submitted
Nov 3, 2020
Last revised
Sep 16, 2024 (v2)
Subjects
math.DG

Comments: 39 pages; simplified some proofs, added details of iteration scheme

Mean curvature flowRicci solitonsSelf-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 rigidity of mean convex self-shrinkers (Qiang Guang and Jonathan J. Zhu, 2016)2Guang 2016 On the entropy of closed hypersurfaces and singular self-shrinkers (Jonathan J. Zhu, 2016)4Zhu 2016 Uniqueness of asymptotically conical tangent flows (Otis Chodosh and Felix Schulze, 2019)7Chodosh 2019 Rigidity and Łojasiewicz inequalities for Clifford self-shrinkers (Ao Sun and Jonathan J. Zhu, 2020)1Sun 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 The PDE-ODI principle and cylindrical mean curvature flows (Richard H. Bamler and Yi Lai, 2025)6Bamler 2025 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. Rigidity and Łojasiewicz inequalities for Clifford self-shrinkers

    Ao Sun and Jonathan J. Zhu

    Topics Self-shrinkers & translators, Stability & uniqueness, Ricci solitons · Same author Jonathan J. Zhu · Shared terms “Łojasiewicz inequalities”, “isolated”

  2. On the rigidity of mean convex self-shrinkers

    Qiang Guang and Jonathan J. ZhuWider flows

    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”, “Colding Minicozzi”

  4. 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 “round cylinder”, “Colding Minicozzi”

  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

  6. The PDE-ODI principle and cylindrical mean curvature flows

    Richard H. Bamler and Yi Lai

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

  7. Uniqueness of asymptotically conical tangent flows

    Otis Chodosh and Felix SchulzeWider flows

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

  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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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)

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  2. 6C.8
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  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

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