Paper · Jul 15, 2020

Convexity estimates for hypersurfaces moving by concave curvature functions

Stephen Lynch

math.DGWider flowsarXiv:2007.07791
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Abstract

We study fully nonlinear geometric flows that deform strictly -convex hypersurfaces in Euclidean space with pointwise normal speed given by a concave function of the principal curvatures. Specifically, the speeds we consider are obtained by performing a nonlinear interpolation between the mean and the -harmonic mean of the principal curvatures. Our main result is a convexity estimate showing that, on compact solutions, regions of high curvature are approximately convex. In contrast to the mean curvature flow, the fully nonlinear flows considered here preserve -convexity in a Riemannian background, and we show that the convexity estimate carries over to this setting as long as the ambient curvature satisfies a natural pinching condition.

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Submitted
Jul 15, 2020
Subjects
math.DG
Mean curvature flowCurvature pinching

Its neighbourhood

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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 Sharp one-sided curvature estimates for mean curvature flow and applications (Mat Langford, 2017)8Langford 2017 Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions (Mat Langford and Stephen Lynch, 2017)1Langford 2017 Pinched Ancient Solutions to the High Codimension Mean Curvature Flow (Stephen Lynch and Huy The Nguyen, 2017)3Lynch 2017 High Codimension Mean Curvature Flow with Surgery (Stephen Lynch and Huy The Nguyen, 2020)4Lynch 2020 Convexity Estimates for High Codimension Mean Curvature Flow (Stephen Lynch and Huy The Nguyen, 2020)7Lynch 2020 Local convexity estimates for mean curvature flow (Mat Langford, 2021)5Langford 2021 Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows (Mat Langford and James McCoy, 2025)6Langford 2025 The planarity estimate in dimensions five and six (Stephen Lynch, 2026)2Lynch 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions

    Mat Langford and Stephen LynchWider flows

    Topics Curvature pinching · Same author Stephen Lynch · Shared terms “fully nonlinear”, “principal curvatures”

  2. The planarity estimate in dimensions five and six

    Stephen LynchWider flows

    Topics Curvature pinching · Same author Stephen Lynch · Shared terms “high curvature”, “principal”

  3. Pinched Ancient Solutions to the High Codimension Mean Curvature Flow

    Stephen Lynch and Huy The NguyenWider flows

    Topics Curvature pinching · Same author Stephen Lynch · Shared terms “pinching condition”

  4. High Codimension Mean Curvature Flow with Surgery

    Stephen Lynch and Huy The NguyenWider flows

    Topics Curvature pinching · Same author Stephen Lynch · Shared terms “high curvature”, “pinching condition”

  5. Local convexity estimates for mean curvature flow

    Mat LangfordWider flows

    Topics Curvature pinching · Shared terms “convexity estimate”

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

    Mat Langford and James McCoyWider flows

    Topics Curvature pinching · Shared terms “fully nonlinear”, “Euclidean space”

  7. Convexity Estimates for High Codimension Mean Curvature Flow

    Stephen Lynch and Huy The NguyenWider flows

    Topics Mean curvature flow · Same author Stephen Lynch · Shared terms “convexity estimate”, “principal”

  8. Sharp one-sided curvature estimates for mean curvature flow and applications

    Mat LangfordWider flows

    Topics Curvature pinching · Shared terms “convexity estimate”, “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)

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.