Paper · Jan 10, 2017

Uniqueness of closed self-similar solutions to -curvature flow

Shanze Gao, Haizhong Li, Hui Ma

math.DGWider flowsarXiv:1701.02642v2
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Abstract

By adapting the test functions introduced by Choi-Daskaspoulos and Brendle-Choi-Daskaspoulos and exploring properties of the -th elementary symmetric functions intensively, we show that for any fixed with , any strictly convex closed hypersurface in satisfying , with , must be a round sphere. In fact, we prove a uniqueness result for any strictly convex closed hypersurface in satisfying , where is a positive homogeneous smooth symmetric function of the principal curvatures and is a constant.

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Submitted
Jan 10, 2017
Last revised
Jan 24, 2017 (v2)
Subjects
math.DG, math.AP

Comments: 23 pages, v2: results improved, references added

Other curvature flowsHomogeneous spacesStability & uniqueness

Its neighbourhood

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  1. Self-similar solutions of -curvature flow

    Shanze Gao 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 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 authors Shanze Gao, Haizhong Li · Shared terms “strictly convex”, “round sphere”, “principal curvatures”

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

    Shanze Gao and Hui MaWider flows

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

  4. Closed self-similar solutions to flows by negative powers of curvature

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

    Kyeongsu Choi and Panagiota DaskalopoulosWider flows

    Topics Other curvature flows, Stability & uniqueness · Shared terms “closed self”, “convex closed”, “strictly convex”

  6. Contracting axially symmetric hypersurfaces by powers of the -curvature

    Haizhong Li, Xianfeng Wang and Jing WuWider flows

    Topics Other curvature flows · Same author Haizhong Li · Shared terms “symmetric function”, “principal curvatures”, “elementary symmetric”

  7. Uniqueness of self-shrinkers to the degree-one curvature flow with a tangent cone at infinity

    Siao-Hao GuoWider flows

    Topics Homogeneous spaces, Other curvature flows, Stability & uniqueness · Shared terms “principal curvatures”

  8. Asymptotic convergence for a class of anisotropic curvature flows

    Haizhong Li, Botong Xu and Ruijia ZhangWider flows

    Topics Other curvature flows · Same author Haizhong Li · Shared terms “elementary symmetric”, “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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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.