Paper · Aug 30, 2017

A survey of closed self-shrinkers with symmetry

Gregory Drugan, Hojoo Lee, Xuan Hien Nguyen

math.DGWider flowsarXiv:1708.09113
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Abstract

In this paper, we survey known results on closed self-shrinkers for mean curvature flow and discuss techniques used in recent constructions of closed self-shrinkers with classical rotational symmetry. We also propose new existence and uniqueness problems for closed self-shrinkers with bi-rotational symmetry and provide numerical evidence for the existence of new examples.

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Submitted
Aug 30, 2017
Subjects
math.DG

Comments: 27 figures

Mean curvature flowSelf-shrinkers & translatorsStability & uniquenessRotational symmetry & warped productsNumerics & 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 Rotational symmetry of self-expanders to the inverse mean curvature flow with cylindrical ends (Gregory Drugan, Frederick Tsz-Ho Fong and Hojoo Lee, 2016)1Drugan 2016 Rotational Symmetry of Asymptotically Conical Mean Curvature Flow Self-Expanders (Frederick Tsz-Ho Fong and Peter McGrath, 2016)8Fong 2016 Shrinking doughnuts via variational methods (Gregory Drugan and Xuan Hien Nguyen, 2017)4Drugan 2017 Numerically computing the index of mean curvature flow self-shrinkers (Yakov Berchenko-Kogan, 2020)5Berchenko-Ko 2020 Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions (Jingze Zhu, 2021)7Zhu 2021 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)3Ketover 2024 A gap Theorem on closed self-shrinkers of mean curvature flow (Yuhang Zhao, 2025)2Zhao 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rotational symmetry of self-expanders to the inverse mean curvature flow with cylindrical ends

    Gregory Drugan, Frederick Tsz-Ho Fong and Hojoo LeeWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Same authors Gregory Drugan, Hojoo Lee

  2. A gap Theorem on closed self-shrinkers of mean curvature flow

    Yuhang ZhaoWider flows

    Topics Self-shrinkers & translators · Shared terms “closed self”, “evidence”

  3. Self-shrinkers whose asymptotic cones fatten

    Daniel KetoverWider flows

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

  4. Shrinking doughnuts via variational methods

    Gregory Drugan and Xuan Hien NguyenWider flows

    Topics Self-shrinkers & translators · Same authors Gregory Drugan, Xuan Hien Nguyen

  5. Numerically computing the index of mean curvature flow self-shrinkers

    Yakov Berchenko-KoganWider flows

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

  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 · Shared terms “numerical evidence”

  7. Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions

    Jingze ZhuWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators

  8. Rotational Symmetry of Asymptotically Conical Mean Curvature Flow Self-Expanders

    Frederick Tsz-Ho Fong and Peter McGrathWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators

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) 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) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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.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.