Paper · Aug 11, 2023

Genus one singularities in mean curvature flow

Adrian Chun-Pong Chu, Ao Sun

math.DGWider flowsarXiv:2308.05923v3Geom. Topol. 29 (2025) 4299-4340
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Abstract

We show that for certain one-parameter families of initial conditions in , when we run mean curvature flow, a genus one singularity must appear in one of the flows. Moreover, such a singularity is robust under perturbation of the family of initial conditions. This contrasts sharply with the case of just a single flow. As an application, we construct an embedded, genus one self-shrinker with entropy lower than a shrinking doughnut.

Comments: 37 pages. Accepted by Geometry & Topology. v3: We added a remark to address the genus one example by Buzano-Nguyen-Schulz and their numerical simulation

Mean curvature flowSelf-shrinkers & translatorsSingularities

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 2021, 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 Noncompact self-shrinkers for mean curvature flow with arbitrary genus (Reto Buzano, Huy The Nguyen and Mario B. Schulz, 2021)5Buzano 2021 Closed embedded self-shrinkers of mean curvature flow (Oskar Riedler, 2022)2Riedler 2022 On mean curvature flow translators with prescribed ends (Ao Sun and Zhihan Wang, 2023)1Sun 2023 Mean curvature flow with generic initial data II (Otis Chodosh, Kyeongsu Choi and Felix Schulze, 2023)4Chodosh 2023 Solitons to Mean Curvature Flow in the hyperbolic 3-space (R. F. de Lima, A. K. Ramos and J. P. dos Santos, 2023)7de Lima 2023 Embedded cylindrical and doughnut-shaped λ-hypersurfaces (Qing-Ming Cheng, Junqi Lai and Guoxin Wei, 2024)3Cheng 2024 Self-expanders of positive genus (Guanhua Shao and Jiahua Zou, 2025)6Shao 2025 Mean curvature flows with cylindrical singularities II: stability and genericity (Ao Sun and Jinxin Xue, 2026)8Sun 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On mean curvature flow translators with prescribed ends

    Ao Sun and Zhihan WangWider flows PDF

    Topics Self-shrinkers & translators · Same author Ao Sun · Shared terms “parameter family”

  2. Closed embedded self-shrinkers of mean curvature flow

    Oskar RiedlerWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “shrinking doughnut”

  3. Embedded cylindrical and doughnut-shaped -hypersurfaces

    Qing-Ming Cheng, Junqi Lai and Guoxin WeiWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “doughnut”

  4. Mean curvature flow with generic initial data II

    Otis Chodosh, Kyeongsu Choi and Felix SchulzeWider flows PDF

    Topics Self-shrinkers & translators, Singularities

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

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

    Topics Self-shrinkers & translators

  6. Self-expanders of positive genus

    Guanhua Shao and Jiahua ZouWider flows PDF

    Topics Self-shrinkers & translators

  7. Solitons to Mean Curvature Flow in the hyperbolic 3-space

    R. F. de Lima, A. K. Ramos and J. P. dos SantosWider flows PDF

    Topics Self-shrinkers & translators · Shared terms “initial condition”, “parameter family”

  8. Mean curvature flows with cylindrical singularities II: stability and genericity

    Ao Sun and Jinxin XueWider flows PDF

    Topics Singularities · Same author Ao Sun · Shared terms “perturbation”

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. 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 usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

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.