Paper · Apr 13, 2021

Asymptotic Convergence for a Class of Fully Nonlinear Contracting Curvature Flows

Yusha Lv, Hejun Wang

math.DGWider flowsarXiv:2104.05966
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Abstract

In this paper, we study a class of fully nonlinear contracting curvature flows of closed, uniformly convex hypersurfaces in the Euclidean space with the normal speed given by or , where is a monotone, symmetric, inverse-concave, homogeneous of degree one function of the principal curvatures, is the distance from the hypersurface to the origin and is the support function of hypersurface. If when or when , we prove that the flow exists for all times and converges to the origin. After proper rescaling, we prove that the normalized flow converges exponentially in the topology to a sphere centered at the origin. Furthermore, for special inverse concave curvature function , where is Gauss curvature and is inverse-concave, we obtain the asymptotic convergence for the flow with when . If , a counterexample is given for the above convergence when speed equals to .

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Submitted
Apr 13, 2021
Subjects
math.DG
Other curvature flowsHomogeneous spacesLong-time existence & convergence

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 Evolution of complete noncompact graphs by powers of curvature function (Guanghan Li and Yusha Lv, 2019)1Li 2019 Contracting self-similar solutions of nonhomogeneous curvature flows (James McCoy, 2020)3McCoy 2020 Asymptotic convergence for modified scalar curvature flow (Ling Xiao, 2020)4Xiao 2020 Self-similar solutions to fully nonlinear curvature flows by high powers of curvature (Shanze Gao, Haizhong Li and Xianfeng Wang, 2020)5Gao 2020 On a Class of Fully Nonlinear Curvature Flows in Hyperbolic Space (Fang Hong, 2021)8Hong 2021 Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone (Ya Gao and Jing Mao, 2024)6Gao 2024 Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows (Weimin Sheng and Jiazhuo Yang, 2025)2Sheng 2025 Noncollapsing for Curvature Flows with Inhomogeneous Speeds (Weimin Sheng and Ye Zhu, 2025)7Sheng 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Evolution of complete noncompact graphs by powers of curvature function

    Guanghan Li and Yusha LvWider flows

    Topics Homogeneous spaces, Other curvature flows · Same author Yusha Lv · Shared terms “inverse concave”, “curvature function”, “concave curvature”

  2. Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows

    Weimin Sheng and Jiazhuo YangWider flows

    Topics Homogeneous spaces, Long-time existence & convergence, Other curvature flows · Shared terms “fully nonlinear”, “flow exists”, “convex hypersurfaces”

  3. Contracting self-similar solutions of nonhomogeneous curvature flows

    James McCoyWider flows

    Topics Homogeneous spaces, Other curvature flows · Shared terms “contracting”, “inverse concave”, “fully nonlinear”

  4. Asymptotic convergence for modified scalar curvature flow

    Ling XiaoWider flows

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “asymptotic convergence”, “sphere centered”, “converges exponentially”

  5. Self-similar solutions to fully nonlinear curvature flows by high powers of curvature

    Shanze Gao, Haizhong Li and Xianfeng WangWider flows

    Topics Homogeneous spaces, Other curvature flows · Shared terms “fully nonlinear”, “inverse concave”, “principal curvatures”

  6. Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone

    Ya Gao and Jing MaoWider flows

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “asymptotic convergence”, “fully nonlinear”, “speed equals”

  7. Noncollapsing for Curvature Flows with Inhomogeneous Speeds

    Weimin Sheng and Ye ZhuWider flows

    Topics Homogeneous spaces, Other curvature flows · Shared terms “inverse concave”, “fully nonlinear”, “monotone”

  8. On a Class of Fully Nonlinear Curvature Flows in Hyperbolic Space

    Fang HongWider flows

    Topics Long-time existence & convergence, Other curvature flows · Shared terms “fully nonlinear”, “uniformly convex”, “principal curvatures”

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.

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) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.