Paper · Jul 25, 2025

Self-expanders of positive genus

Guanhua Shao, Jiahua Zou

math.DGWider flowsarXiv:2507.19428v2
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Abstract

For a general class of cones in , we construct self-expanders of positive genus asymptotic to these cones. As a result, we use these self-expanders to construct a mean curvature flow with genus strictly decreasing but not to zero at the first singular time. We also construct a sequence of self-expanders with unbounded genus which are asymptotic to the same rotationally symmetric cone. Moreover, we characterize the asymptotic behavior of the sequence.

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Submitted
Jul 25, 2025
Last revised
Aug 7, 2025 (v2)
Subjects
math.DG

Comments: 25 pages, all comments are welcome

Mean curvature flowSelf-shrinkers & translatorsWeak & singular flowsRotational symmetry & warped products

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 Rotational symmetry of solutions of mean curvature flow coming out of a double cone II (Letian Chen, 2022)7Chen 2022 Entire self-expanders for power of σₖ curvature flow in Minkowski space (Zhizhang Wang and Ling Xiao, 2022)4Wang 2022 Self-expanders to the mean curvature flow based on the generalized Lawson-Osserman cone (Chen-Kuan Lee, 2022)2Lee 2022 Genus one singularities in mean curvature flow (Adrian Chun-Pong Chu and Ao Sun, 2023)6Chu 2023 Minimal hypersurfaces in S⁴(1) by doubling the equatorial S³ (Nikolaos Kapouleas and Jiahua Zou, 2024)8Kapouleas 2024 Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow (Junming Xie, 2025)3Xie 2025 The rigidity theorems for complete self-expander of mean curvature flow (Zhi Li and Guoxin Wei, 2026)5Li 2026 On knotted self expanders of the mean curvature flow in R³ (Alexander Mramor and Shengwen Wang, 2026)1Mramor 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On knotted self expanders of the mean curvature flow in

    Alexander Mramor and Shengwen Wang

    Topics Self-shrinkers & translators · Shared terms “self expanders”

  2. Self-expanders to the mean curvature flow based on the generalized Lawson-Osserman cone

    Chen-Kuan LeeWider flows

    Topics Self-shrinkers & translators · Shared terms “self expanders”

  3. Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow

    Junming XieWider flows

    Topics Self-shrinkers & translators · Shared terms “self expanders”

  4. Entire self-expanders for power of curvature flow in Minkowski space

    Zhizhang Wang and Ling XiaoWider flows

    Topics Self-shrinkers & translators · Shared terms “self expanders”

  5. The rigidity theorems for complete self-expander of mean curvature flow

    Zhi Li and Guoxin WeiWider flows

    Topics Self-shrinkers & translators · Shared terms “self expanders”

  6. Genus one singularities in mean curvature flow

    Adrian Chun-Pong Chu and Ao SunWider flows

    Topics Self-shrinkers & translators

  7. Rotational symmetry of solutions of mean curvature flow coming out of a double cone II

    Letian ChenWider flows

    Topics Rotational symmetry & warped products, Weak & singular flows · Shared terms “rotationally symmetric”

  8. Minimal hypersurfaces in by doubling the equatorial

    Nikolaos Kapouleas and Jiahua ZouWider flows

    Topics Self-shrinkers & translators, Weak & singular flows · Same author Jiahua Zou

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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002) 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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  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.