Paper · Apr 13, 2020

Bounds on the Index of Rotationally Symmetric Self-Shrinking Tori

Yakov Berchenko-Kogan

math.DGWider flowsarXiv:2004.06217v2
View PDF View HTML arXiv page

As on arXiv

Abstract

A closed surface evolving under mean curvature flow becomes singular in finite time. Near the singularity, the surface resembles a self-shrinker, a surface that shrinks by dilations under mean curvature flow. If the singularity is modeled on a self-shrinker other than a round sphere or cylinder, then the singularity is unstable under perturbations of the flow. One can quantify this instability using the index of the self-shrinker when viewed as a critical point of the entropy functional. In this work, we prove an upper bound on the index of rotationally symmetric self-shrinking tori in terms of their entropy and their maximum and minimum radii. While there have been a few lower bound results in the literature, we believe that this result is the first upper bound on the index of a self-shrinker. Our methods also give lower bounds on the index and the entropy, and our methods give simple formulas for two entropy-decreasing variations whose existence was proved by Liu. Surprisingly, the eigenvalue corresponding to these variations is exactly . Finally, we present some preliminary results in higher dimensions and six potential directions for future work.

View PDF View HTML arXiv page
Submitted
Apr 13, 2020
Last revised
Sep 22, 2020 (v2)
Subjects
math.DG
Journal
Geometriae Dedicata (2021) 213:83-106
DOI
10.1007/s10711-020-00569-9

Comments: Minor updates and corrections, a shortened abstract, and a DOI reference to the published version. 27 pages, 2 figures

Mean curvature flowSelf-shrinkers & translatorsSingularitiesStability & uniquenessRotational symmetry & warped productsEigenvalues & spectrum

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 The index of shrinkers of the mean curvature flow (Zihan Hans Liu, 2016)5Liu 2016 Geometry of Lagrangian self-shrinking tori and applications to the Piecewise Lagrangian Mean Curvature Flow (Jingyi Chen and John Man Shun Ma, 2016)4Chen 2016 The entropy of the Angenent torus is approximately 1.85122 (Yakov Berchenko-Kogan, 2018)3Berchenko-Ko 2018 Symmetric self-shrinkers for the fractional mean curvature flow (Annalisa Cesaroni and Matteo Novaga, 2018)7Cesaroni 2018 Numerically computing the index of mean curvature flow self-shrinkers (Yakov Berchenko-Kogan, 2020)1Berchenko-Ko 2020 Comparison Theorems for the Profile Curve Equation of Rotationally Symmetric Self-Shrinkers (Peng Peng, 2026)2Peng 2026 The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow (Fagui Li and Yuhang Zhao, 2026)6Li 2026 Singularity formation and self-shrinkers for the parabolic Bernoulli problem (Hunter Liu, Sebastian Munoz and Zihui Zhao, 2026)8Liu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Numerically computing the index of mean curvature flow self-shrinkers

    Yakov Berchenko-KoganWider flows

    Topics Eigenvalues & spectrum, Rotational symmetry & warped products, Self-shrinkers & translators · Same author Yakov Berchenko-Kogan · Shared terms “symmetric self”, “dilations”, “rotationally symmetric”

  2. Comparison Theorems for the Profile Curve Equation of Rotationally Symmetric Self-Shrinkers

    Peng PengWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators, Singularities · Shared terms “symmetric self”, “rotationally symmetric”, “self shrinking”

  3. The entropy of the Angenent torus is approximately 1.85122

    Yakov Berchenko-KoganWider flows

    Topics Self-shrinkers & translators, Singularities · Same author Yakov Berchenko-Kogan · Shared terms “dilations”

  4. Geometry of Lagrangian self-shrinking tori and applications to the Piecewise Lagrangian Mean Curvature Flow

    Jingyi Chen and John Man Shun MaWider flows

    Topics Self-shrinkers & translators, Singularities · Shared terms “shrinking tori”, “self shrinking”, “upper bound”

  5. The index of shrinkers of the mean curvature flow

    Zihan Hans Liu

    Topics Rotational symmetry & warped products · Shared terms “rotationally symmetric”, “higher dimensions”

  6. The Rigidity Theorems for Self-Shrinkers in the Mean Curvature Flow

    Fagui Li and Yuhang ZhaoWider flows

    Topics Eigenvalues & spectrum, Self-shrinkers & translators, Stability & uniqueness · Shared terms “self shrinking”

  7. Symmetric self-shrinkers for the fractional mean curvature flow

    Annalisa Cesaroni and Matteo NovagaWider flows

    Topics Self-shrinkers & translators, Stability & uniqueness · Shared terms “symmetric self”, “unstable”

  8. Singularity formation and self-shrinkers for the parabolic Bernoulli problem

    Hunter Liu, Sebastian Munoz and Zihui ZhaoWider flows

    Topics Eigenvalues & spectrum, Self-shrinkers & translators, Singularities · Shared terms “self shrinking”, “unstable”

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

  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.