Paper · Nov 3, 2020

Rigidity and Łojasiewicz inequalities for Clifford self-shrinkers

Ao Sun, Jonathan J. Zhu

math.DGarXiv:2011.01636
View PDF View HTML arXiv page

As on arXiv

Abstract

We show that the product of two round shrinking spheres is an isolated self-shrinker in any codimension, modulo rotations. Moreover we prove explicit Łojasiewicz inequalities near such products. Łojasiewicz inequalities were previously used by Schulze to prove uniqueness of tangent mean curvature flows at compact shrinkers; our results provide an explicit rate of convergence to products of two spheres.

View PDF View HTML arXiv page
Submitted
Nov 3, 2020
Subjects
math.DG

Comments: 31 pages, comments welcome!

Mean curvature flowRicci solitonsSelf-shrinkers & translatorsStability & uniqueness

Its neighbourhood

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)4Guang 2016 On the entropy of closed hypersurfaces and singular self-shrinkers (Jonathan J. Zhu, 2016)6Zhu 2016 Local Entropy and Generic Multiplicity One Singularities of Mean Curvature Flow of Surfaces (Ao Sun, 2018)8Sun 2018 Bifurcation of perturbations of non-generic closed self-shrinkers (Zhengjiang Lin and Ao Sun, 2020)7Lin 2020 Łojasiewicz inequalities, uniqueness and rigidity for cylindrical self-shrinkers (Jonathan J. Zhu, 2020)1Zhu 2020 Rigidity of spherical product Ricci solitons (Ao Sun and Jonathan J. Zhu, 2021)2Sun 2021 Uniqueness of blowups for forced mean curvature flow (Sven Hirsch and Jonathan J. Zhu, 2023)5Hirsch 2023 A stable self-shrinking Möbius bundle in ℝ⁴ (Kahnrad Braxton, Tang-Kai Lee and Jonathan J. Zhu, 2026)3Braxton 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Łojasiewicz inequalities, uniqueness and rigidity for cylindrical self-shrinkers

    Jonathan J. Zhu

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

  2. Rigidity of spherical product Ricci solitons

    Ao Sun and Jonathan J. Zhu

    Topics Stability & uniqueness, Ricci solitons · Same authors Ao Sun, Jonathan J. Zhu · Shared terms “Clifford”, “isolated”

  3. A stable self-shrinking Möbius bundle in

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

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

  4. On the rigidity of mean convex self-shrinkers

    Qiang Guang and Jonathan J. ZhuWider flows

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

  5. Uniqueness of blowups for forced mean curvature flow

    Sven Hirsch and Jonathan J. ZhuWider flows

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

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

    Jonathan J. ZhuWider flows

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

  7. Bifurcation of perturbations of non-generic closed self-shrinkers

    Zhengjiang Lin and Ao SunWider flows

    Topics Self-shrinkers & translators · Same author Ao Sun

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

    Ao SunWider flows

    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) 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) 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

  1. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  3. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.