Paper · Dec 6, 2018

Uniqueness Theorems of Self-Conformal Solutions to Inverse Curvature Flows

Nicholas Cheng-Hoong Chin, Frederick Tsz-Ho Fong, Jingbo Wan

math.DGWider flowsarXiv:1812.02396v2
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Abstract

It has been known in that round spheres are the only closed homothetic self-similar solutions to the inverse mean curvature flow and parabolic curvature flows by degree -1 homogeneous functions of principle curvatures in the Euclidean space. In this article, we prove that the round sphere is rigid in much stronger sense, that under some natural conditions such as star-shapedness, it is the only closed solution to the inverse mean curavture flow and the above-mentioned flows in the Euclidean space which evolves by diffeomorphisms generated by conformal Killing fields.

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Submitted
Dec 6, 2018
Last revised
Apr 28, 2020 (v2)
Subjects
math.DG

Comments: 15 pages; Theorems 3.1 and 4.5 rewritten; some remarks added. To appear in Proc. Amer. Math. Soc

Mean curvature flowOther curvature flowsInverse mean curvature flowHomogeneous spacesStability & uniqueness

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)4Drugan 2016 Non-parametric inverse curvature flows in the AdS-Schwarzschild manifold (Li Chen and Jing Mao, 2016)7Chen 2016 Self-Expanders to Inverse Curvature Flows by Homogeneous Functions (Tsz-Kiu Aaron Chow, Ka-Wing Chow and Frederick Tsz-Ho Fong, 2017)1Chow 2017 Self similar solutions to the inverse mean curvature flow in ℝ² (Jui-En Chang, 2018)3Chang 2018 Shifted inverse curvature flows in hyperbolic space (Xianfeng Wang, Yong Wei and Tailong Zhou, 2020)5Wang 2020 A class of inverse curvature flows for star-shaped hypersurfaces evolving in a cone (Jing Mao and Qiang Tu, 2021)8Mao 2021 Classification of ruled surfaces as homothetic self-similar solutions of the inverse mean curvature flow in the Lorentz-Minkowski 3-space (Gregório Silva Neto and Vanessa Silva, 2023)6Neto 2023 Uniqueness and Symmetry of Self-Similar Solutions of Curvature Flows in Warped Product Spaces (Frederick Tsz-Ho Fong, 2024)2Fong 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Self-Expanders to Inverse Curvature Flows by Homogeneous Functions

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

    Topics Inverse mean curvature flow, Homogeneous spaces, Other curvature flows · Same author Frederick Tsz-Ho Fong · Shared terms “homogeneous functions”, “round sphere”, “Euclidean space”

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

    Frederick Tsz-Ho FongWider flows

    Topics Inverse mean curvature flow, Homogeneous spaces, Other curvature flows · Same author Frederick Tsz-Ho Fong

  3. Self similar solutions to the inverse mean curvature flow in

    Jui-En ChangWider flows

    Topics Inverse mean curvature flow, Other curvature flows

  4. 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 Inverse mean curvature flow, Other curvature flows · Same author Frederick Tsz-Ho Fong

  5. Shifted inverse curvature flows in hyperbolic space

    Xianfeng Wang, Yong Wei and Tailong ZhouWider flows

    Topics Inverse mean curvature flow, Homogeneous spaces, Other curvature flows

  6. Classification of ruled surfaces as homothetic self-similar solutions of the inverse mean curvature flow in the Lorentz-Minkowski 3-space

    Gregório Silva Neto and Vanessa SilvaWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “homothetic self”

  7. Non-parametric inverse curvature flows in the AdS-Schwarzschild manifold

    Li Chen and Jing MaoWider flows

    Topics Inverse mean curvature flow, Homogeneous spaces, Other curvature flows · Shared terms “principle curvatures”

  8. A class of inverse curvature flows for star-shaped hypersurfaces evolving in a cone

    Jing Mao and Qiang TuWider flows

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “round sphere”, “Euclidean”

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) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  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.