Paper · Nov 23, 2016

Self-similar solutions of -curvature flow

Shanze Gao, Hui Ma

math.DGWider flowsarXiv:1611.07597
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Abstract

In this paper, employing a new inequality, we show that under certain curvature pinching condition, the strictly convex closed smooth self-similar solution of -flow must be a round sphere. We also obtain a similar result for the solutions of with a non-homogeneous function . At last, we prove that if can be compared with , then a closed strictly -convex solution of must be a round sphere.

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Submitted
Nov 23, 2016
Subjects
math.DG, math.AP

Comments: 14 pages

Other curvature flowsCurvature pinchingHomogeneous spaces

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 Uniqueness of closed self-similar solutions to the Gauss curvature flow (Kyeongsu Choi and Panagiota Daskalopoulos, 2016)4Choi 2016 Mixed volume preserving flow by powers of homogeneous curvature functions of degree one (Shunzi Guo, 2016)6Guo 2016 Uniqueness of closed self-similar solutions to σₖ^α-curvature flow (Shanze Gao, Haizhong Li and Hui Ma, 2017)1Gao 2017 Self-similar solutions of curvature flows in warped products (Shanze Gao and Hui Ma, 2018)2Gao 2018 A Pinching Estimate for Convex Hypersurfaces Evolving Under a Nonhomogeneous Variant of Mean Curvature Flow (Tim Espin, 2020)8Espin 2020 Self-similar solutions to fully nonlinear curvature flows by high powers of curvature (Shanze Gao, Haizhong Li and Xianfeng Wang, 2020)3Gao 2020 Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows (Mat Langford and James McCoy, 2025)5Langford 2025 Entropy and round-point convergence for the σₖ-curvature flow (Tianci Luo, Yong Wei, Ruijia Zhang et al., 2026)7Luo 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniqueness of closed self-similar solutions to -curvature flow

    Shanze Gao, Haizhong Li and Hui MaWider flows

    Topics Homogeneous spaces, Other curvature flows · Same authors Shanze Gao, Hui Ma · Shared terms “convex closed”, “round sphere”, “strictly convex”

  2. Self-similar solutions of curvature flows in warped products

    Shanze Gao and Hui MaWider flows

    Topics Other curvature flows · Same authors Shanze Gao, Hui Ma · Shared terms “closed strictly”, “strictly convex”

  3. 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 · Same author Shanze Gao · Shared terms “closed strictly”, “round sphere”, “strictly convex”

  4. Uniqueness of closed self-similar solutions to the Gauss curvature flow

    Kyeongsu Choi and Panagiota DaskalopoulosWider flows

    Topics Other curvature flows · Shared terms “closed smooth”, “convex closed”, “smooth self”

  5. Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows

    Mat Langford and James McCoyWider flows

    Topics Curvature pinching, Homogeneous spaces, Other curvature flows · Shared terms “homogeneous functions”

  6. Mixed volume preserving flow by powers of homogeneous curvature functions of degree one

    Shunzi GuoWider flows

    Topics Curvature pinching, Homogeneous spaces, Other curvature flows · Shared terms “pinching condition”, “round sphere”

  7. Entropy and round-point convergence for the -curvature flow

    Tianci Luo, Yong Wei, Ruijia Zhang et al.Wider flows

    Topics Curvature pinching, Other curvature flows · Shared terms “closed strictly”, “strictly convex”

  8. A Pinching Estimate for Convex Hypersurfaces Evolving Under a Nonhomogeneous Variant of Mean Curvature Flow

    Tim EspinWider flows

    Topics Curvature pinching, Homogeneous spaces · Shared terms “pinching condition”

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) Curvature pinching Curvature conditions that Ricci flow preserves and improves. Hamilton–Ivey pinching makes positive curvature dominate in 3D blow-ups, and invariant… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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. 6A.9
    Reaction–Diffusion: Fronts, Patterns and PinchingBook 6A · The Heat Equation and Its Relatives

    Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.

  2. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.

  3. 11A.10
    Hamilton–Ivey PinchingBook 11A · The Ricci Flow: Existence and Maximum Principles

    Why high curvature in dimension 3 is almost positive.