Paper · Nov 9, 2020

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

Shanze Gao, Haizhong Li, Xianfeng Wang

math.DGWider flowsarXiv:2011.04200v2
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Abstract

In this paper, we investigate closed strictly convex hypersurfaces in which shrink self-similarly under a large family of fully nonlinear curvature flows by high powers of curvature. When the speed function is given by powers of a homogeneous of degree and inverse concave function of the principal curvatures with power greater than , we prove that the only such hypersurfaces are round spheres. We also prove that slices are the only closed strictly convex self-similar solutions to such curvature flows in the hemisphere with power greater than or equal to .

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Submitted
Nov 9, 2020
Last revised
Sep 27, 2021 (v2)
Subjects
math.DG

Comments: 23 pages, revised version, some typos are corrected, to appear in Crelle's Journal

Other curvature flowsHomogeneous spaces

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    Shanze Gao, Haizhong Li and Hui MaWider flows

    Topics Homogeneous spaces, Other curvature flows · Same authors Shanze Gao, Haizhong Li · Shared terms “strictly convex”, “round sphere”, “principal curvatures”

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    Shanze Gao and Hui MaWider flows

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

  3. Non-homothetic convex ancient solutions for flows by high powers of curvature

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    Topics Homogeneous spaces, Other curvature flows · Shared terms “high powers”, “speed function”, “principal curvatures”

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    Haizhong Li, Xianfeng Wang and Jing WuWider flows

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

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    Topics Homogeneous spaces, Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”, “convex hypersurfaces”

  6. Contracting self-similar solutions of nonhomogeneous curvature flows

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

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

    Shanze Gao and Hui MaWider flows

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  8. Asymptotic Convergence for a Class of Fully Nonlinear Contracting Curvature Flows

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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)

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.