Paper · Jul 12, 2020

Numerically computing the index of mean curvature flow self-shrinkers

Yakov Berchenko-Kogan

math.DGWider flowsarXiv:2007.06094
View PDF View HTML arXiv page

As on arXiv

Abstract

Surfaces that evolve by mean curvature flow develop singularities. These singularities can be modeled by self-shrinkers, surfaces that shrink by dilations under the flow. Singularities modeled on classical self-shrinkers, namely spheres and cylinders, are stable under perturbations of the flow. In contrast, singularities modeled on other self-shrinkers, such as the Angenent torus, are unstable: perturbing the flow will generally change the kind of singularity. One can measure the degree of instability by computing the Morse index of the self-shrinker, viewed as a critical point of an appropriate functional. In this paper, we present a numerical method for computing the index of rotationally symmetric self-shrinkers. We apply this method to the Angenent torus, the first known nontrivial example of a self-shrinker. We find that, excluding dilations and translations, the index of the Angenent torus is , which is consistent with the lower bound of from the work of Liu and the upper bound of from our earlier work. Also, we unexpectedly discover two additional variations of the Angenent torus with eigenvalue .

View PDF View HTML arXiv page
Submitted
Jul 12, 2020
Subjects
math.DG, math.NA

Comments: 27 pages, 10 figures, 1 table, 1 appendix

Mean curvature flowSelf-shrinkers & translatorsSingularitiesStability & uniquenessRotational symmetry & warped productsEigenvalues & spectrumNumerics & 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 A survey of closed self-shrinkers with symmetry (Gregory Drugan, Hojoo Lee and Xuan Hien Nguyen, 2017)4Drugan 2017 The entropy of the Angenent torus is approximately 1.85122 (Yakov Berchenko-Kogan, 2018)1Berchenko-Ko 2018 Bounds on the Index of Rotationally Symmetric Self-Shrinking Tori (Yakov Berchenko-Kogan, 2020)2Berchenko-Ko 2020 Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds (Harald Garcke and Robert Nürnberg, 2020)8Garcke 2020 A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup (David Garfinkle, James Isenberg, Dan Knopf et al., 2021)7Garfinkle 2021 A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II (David Garfinkle, James Isenberg, Dan Knopf et al., 2022)6Garfinkle 2022 Self-shrinkers whose asymptotic cones fatten (Daniel Ketover, 2024)5Ketover 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. The entropy of the Angenent torus is approximately 1.85122

    Yakov Berchenko-KoganWider flows

    Topics Numerics & visualization, Self-shrinkers & translators, Singularities · Same author Yakov Berchenko-Kogan · Shared terms “Angenent torus”, “numerically”, “nontrivial examples”

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

    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”

  3. 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 “Angenent torus”, “symmetric self”, “rotationally symmetric”

  4. A survey of closed self-shrinkers with symmetry

    Gregory Drugan, Hojoo Lee and Xuan Hien NguyenWider flows

    Topics Numerics & visualization, Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “numerical”

  5. Self-shrinkers whose asymptotic cones fatten

    Daniel KetoverWider flows

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

  6. A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II

    David Garfinkle, James Isenberg, Dan Knopf et al.Wider flows

    Topics Numerics & visualization, Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “numerical”, “rotationally symmetric”, “perturbation”

  7. A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup

    David Garfinkle, James Isenberg, Dan Knopf et al.Wider flows

    Topics Numerics & visualization, Rotational symmetry & warped products, Singularities · Shared terms “numerically”, “rotationally symmetric”, “perturbation”

  8. Numerical approximation of boundary value problems for curvature flow and elastic flow in Riemannian manifolds

    Harald Garcke and Robert NürnbergWider flows

    Topics Numerics & visualization, Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “symmetric self”, “numerically”, “Angenent”

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. 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.