Paper · Aug 24, 2018

The entropy of the Angenent torus is approximately 1.85122

Yakov Berchenko-Kogan

math.DGWider flowsarXiv:1808.08163v2
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Abstract

To study the singularities that appear in mean curvature flow, one must understand self-shrinkers, surfaces that shrink by dilations under mean curvature flow. The simplest examples of self-shrinkers are spheres and cylinders. In 1989, Angenent constructed the first nontrivial example of a self-shrinker, a torus. A key quantity in the study of the formation of singularities is the entropy, defined by Colding and Minicozzi based on work of Huisken. The values of the entropy of spheres and cylinders have explicit formulas, but there is no known formula for the entropy of the Angenent torus. In this work, we numerically estimate the entropy of the Angenent torus using the discrete Euler-Lagrange equations.

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Submitted
Aug 24, 2018
Last revised
Aug 5, 2019 (v2)
Subjects
math.DG, math.NA
DOI
10.1080/10586458.2019.1583616

Comments: 14 pages, 5 figures, Jupyter code included as ancillary file. Journal of Experimental Mathematics, 2019

Mean curvature flowSelf-shrinkers & translatorsSingularitiesNumerics & visualization

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)8Guang 2016 On the entropy of closed hypersurfaces and singular self-shrinkers (Jonathan J. Zhu, 2016)7Zhu 2016 A survey of closed self-shrinkers with symmetry (Gregory Drugan, Hojoo Lee and Xuan Hien Nguyen, 2017)5Drugan 2017 Bounds on the Index of Rotationally Symmetric Self-Shrinking Tori (Yakov Berchenko-Kogan, 2020)4Berchenko-Ko 2020 Numerically computing the index of mean curvature flow self-shrinkers (Yakov Berchenko-Kogan, 2020)1Berchenko-Ko 2020 Noncompact self-shrinkers for mean curvature flow with arbitrary genus (Reto Buzano, Huy The Nguyen and Mario B. Schulz, 2021)6Buzano 2021 Self-shrinkers whose asymptotic cones fatten (Daniel Ketover, 2024)2Ketover 2024 Comparison Theorems for the Profile Curve Equation of Rotationally Symmetric Self-Shrinkers (Peng Peng, 2026)3Peng 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 Numerics & visualization, Self-shrinkers & translators, Singularities · Same author Yakov Berchenko-Kogan · Shared terms “Angenent torus”, “numerically”, “nontrivial examples”

  2. Self-shrinkers whose asymptotic cones fatten

    Daniel KetoverWider flows

    Topics Numerics & visualization, Self-shrinkers & translators · Shared terms “Angenent”

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

    Peng PengWider flows

    Topics Self-shrinkers & translators, Singularities · Shared terms “Angenent torus”, “nontrivial examples”

  4. Bounds on the Index of Rotationally Symmetric Self-Shrinking Tori

    Yakov Berchenko-KoganWider flows

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

  5. A survey of closed self-shrinkers with symmetry

    Gregory Drugan, Hojoo Lee and Xuan Hien NguyenWider flows

    Topics Numerics & visualization, Self-shrinkers & translators

  6. Noncompact self-shrinkers for mean curvature flow with arbitrary genus

    Reto Buzano, Huy The Nguyen and Mario B. SchulzWider flows

    Topics Numerics & visualization, Self-shrinkers & translators

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

    Jonathan J. ZhuWider flows

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

  8. On the rigidity of mean convex self-shrinkers

    Qiang Guang and Jonathan J. ZhuWider flows

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

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) Numerics & visualization Computing and visualising Ricci flow: simulations of neckpinches and critical behaviour, pictures of rotationally symmetric flows, and… Start with: Deforming metrics in the direction of their Ricci tensors (DeTurck, 1983)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  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.