Paper · Nov 30, 2023

Pinching estimates of hypersurfaces by a generalized Gauss curvature flow

Jinrong Hu, Ping Zhang

math.APWider flowsarXiv:2311.18606
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Abstract

A variant of the Gauss curvature flow for closed and convex hypersurfaces is considered. We reveal that if the initial hypersurface is pinched enough, then this property is preserved. Furthermore, based on some structure assumptions on the speed function of the shrinking flow, we show that the flow converges to a sphere. This may generalize the result of B. Chow to the possible non-homogeneous curvature flows.

Comments: This article was written in 2022 and still under review, we forgot to post it on arxiv

Other curvature flowsGauss curvature flowCurvature pinchingHomogeneous spaces

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  1. The chord Gauss curvature flow and its chord Minkowski problem

    Jinrong Hu, Yong Huang, Jian Lu et al.Wider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Same author Jinrong Hu

  2. Volume preserving nonhomogeneous Gauss curvature flow in hyperbolic space

    Yong Wei, Bo Yang and Tailong ZhouWider flows PDF

    Topics Gauss curvature flow, Homogeneous spaces, Other curvature flows · Shared terms “speed function”, “convex hypersurfaces”

  3. Regularity of the -Gauss curvature flow with flat side

    Genggeng Huang, Xu-Jia Wang and Yang ZhouWider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Shared terms “convex hypersurfaces”

  4. Some remarks on a class of logarithmic curvature flow

    Jinrong Hu and Qiongfang MaoWider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Same author Jinrong Hu

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

    Mat Langford and James McCoyWider flows PDF

    Topics Curvature pinching, Homogeneous spaces, Other curvature flows · Shared terms “pinching estimate”, “speed function”

  6. Volume preserving Gauss curvature flow of convex hypersurfaces in the hyperbolic space

    Yong Wei, Bo Yang and Tailong ZhouWider flows PDF

    Topics Gauss curvature flow, Other curvature flows · Shared terms “convex hypersurfaces”, “initial hypersurface”, “flow converges”

  7. A Gauss curvature flow related to the Orlicz-Aleksandrov problem

    Bin Chen and Peibiao ZhaoWider flows PDF

    Topics Gauss curvature flow, Other curvature flows

  8. Invariant -translators for the Gauss curvature flow in Euclidean space

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    Topics Gauss curvature flow, Other curvature flows

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) Gauss curvature flow Wider flows Convex hypersurfaces shrink with normal speed a power K^α of their Gauss curvature. Firey proposed it as a model of stones worn by waves; for α>… 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.