Paper · Jan 8, 2020

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

Tim Espin

math.APWider flowsarXiv:2001.02546
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Abstract

We study a variant of the mean curvature flow for closed, convex hypersurfaces where the normal velocity is a nonhomogeneous function of the principal curvatures. We show that if the initial hypersurface satisfies a certain pinching condition, then this is preserved and the flow converges to a sphere under rescaling.

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Submitted
Jan 8, 2020
Subjects
math.AP

Comments: 20 Pages

Mean curvature flowCurvature 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 Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes (Giuseppe Pipoli and Carlo Sinestrari, 2016)8Pipoli 2016 Mixed volume preserving flow by powers of homogeneous curvature functions of degree one (Shunzi Guo, 2016)6Guo 2016 Self-similar solutions of σₖ^α-curvature flow (Shanze Gao and Hui Ma, 2016)5Gao 2016 Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions (Mat Langford and Stephen Lynch, 2017)4Langford 2017 New pinching estimates for Inverse curvature flows in space forms (Yong Wei, 2017)3Wei 2017 Convexity estimates for hypersurfaces moving by concave curvature functions (Stephen Lynch, 2020)7Lynch 2020 Pinching estimates of hypersurfaces by a generalized Gauss curvature flow (Jinrong Hu and Ping Zhang, 2023)1Hu 2023 Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows (Mat Langford and James McCoy, 2025)2Langford 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Pinching estimates of hypersurfaces by a generalized Gauss curvature flow

    Jinrong Hu and Ping ZhangWider flows

    Topics Curvature pinching, Homogeneous spaces · Shared terms “pinching estimate”, “convex hypersurfaces”, “initial hypersurface”

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

    Mat Langford and James McCoyWider flows

    Topics Curvature pinching, Homogeneous spaces · Shared terms “hypersurfaces evolving”, “pinching estimate”

  3. New pinching estimates for Inverse curvature flows in space forms

    Yong WeiWider flows

    Topics Curvature pinching, Homogeneous spaces · Shared terms “pinching estimate”, “convex hypersurfaces”, “principal curvatures”

  4. Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions

    Mat Langford and Stephen LynchWider flows

    Topics Curvature pinching, Homogeneous spaces · Shared terms “hypersurfaces evolving”, “pinching estimate”, “principal curvatures”

  5. Self-similar solutions of -curvature flow

    Shanze Gao and Hui MaWider flows

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

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

    Shunzi GuoWider flows

    Topics Curvature pinching, Homogeneous spaces · Shared terms “convex hypersurfaces”, “hypersurface satisfies”, “pinching condition”

  7. Convexity estimates for hypersurfaces moving by concave curvature functions

    Stephen LynchWider flows

    Topics Curvature pinching · Shared terms “convex hypersurfaces”, “principal curvatures”, “pinching condition”

  8. Cylindrical estimates for mean curvature flow of hypersurfaces in CROSSes

    Giuseppe Pipoli and Carlo SinestrariWider flows

    Topics Curvature pinching · Shared terms “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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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.