Paper · Sep 7, 2016

Rotational Symmetry of Asymptotically Conical Mean Curvature Flow Self-Expanders

Frederick Tsz-Ho Fong, Peter McGrath

math.DGWider flowsarXiv:1609.02105
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Abstract

In this article, we examine complete, mean-convex self-expanders for the mean curvature flow whose ends have decaying principal curvatures. We prove a Liouville-type theorem associated to this class of self-expanders. As an application, we show that mean-convex self-expanders which are asymptotic to -invariant cones are rotationally symmetric.

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Submitted
Sep 7, 2016
Subjects
math.DG, math.AP

Comments: 14 pages. Comments are welcome!

Mean curvature flowSelf-shrinkers & translatorsBoundaries & non-compactRotational 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 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 Self-Expanders to Inverse Curvature Flows by Homogeneous Functions (Tsz-Kiu Aaron Chow, Ka-Wing Chow and Frederick Tsz-Ho Fong, 2017)2Chow 2017 The space of asymptotically conical self-expanders of mean curvature flow (Jacob Bernstein and Lu Wang, 2017)5Bernstein 2017 Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flow (Jacob Bernstein and Lu Wang, 2018)8Bernstein 2018 Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders (Ilyas Khan, 2020)4Khan 2020 Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces (Frederick Tsz-Ho Fong, 2024)6Fong 2024 Self-expanders of positive genus (Guanhua Shao and Jiahua Zou, 2025)7Shao 2025 Convexity of mean convex asymptotically conical self-expanders to the mean curvature flow (Junming Xie, 2025)3Xie 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 author Frederick Tsz-Ho Fong · Shared terms “self expanders”, “rotationally symmetric”

  2. Self-Expanders to Inverse Curvature Flows by Homogeneous Functions

    Tsz-Kiu Aaron Chow, Ka-Wing Chow and Frederick Tsz-Ho FongWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators, Boundaries & non-compact · Same author Frederick Tsz-Ho Fong · Shared terms “self expanders”, “rotationally symmetric”, “principal curvatures”

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

    Junming XieWider flows

    Topics Self-shrinkers & translators, Boundaries & non-compact · Shared terms “self expanders”, “convex self”, “asymptotically conical”

  4. Uniqueness of Asymptotically Conical Higher Codimension Self-Shrinkers and Self-Expanders

    Ilyas KhanWider flows

    Topics Self-shrinkers & translators, Boundaries & non-compact · Shared terms “self expanders”, “asymptotically conical”

  5. The space of asymptotically conical self-expanders of mean curvature flow

    Jacob Bernstein and Lu WangWider flows

    Topics Self-shrinkers & translators, Boundaries & non-compact · Shared terms “self expanders”, “asymptotically conical”

  6. Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces

    Frederick Tsz-Ho FongWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators, Boundaries & non-compact · Same author Frederick Tsz-Ho Fong · Shared terms “self expanders”, “principal curvatures”, “rotationally symmetric”

  7. Self-expanders of positive genus

    Guanhua Shao and Jiahua ZouWider flows

    Topics Rotational symmetry & warped products, Self-shrinkers & translators · Shared terms “self expanders”, “rotationally symmetric”

  8. Smooth compactness for spaces of asymptotically conical self-expanders of mean curvature flow

    Jacob Bernstein and Lu WangWider flows

    Topics Self-shrinkers & translators, Boundaries & non-compact · Shared terms “self expanders”, “asymptotically conical”, “convex self”

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) 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. 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.