Paper · Apr 12, 2017

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

Mat Langford, Stephen Lynch

math.DGWider flowsarXiv:1704.03802
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Abstract

We prove several sharp one-sided pinching estimates for immersed and embedded hypersurfaces evolving by various fully nonlinear, one-homogeneous curvature flows by the method of Stampacchia iteration. These include sharp estimates for the largest principal curvature and the inscribed curvature ('cylindrical estimates') for flows by concave speeds and a sharp estimate for the exscribed curvature for flows by convex speeds. Making use of a recent idea of Huisken and Sinestrari, we then obtain corresponding estimates for ancient solutions. In particular, this leads to various characterisations of the shrinking sphere amongst ancient solutions of these flows.

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Submitted
Apr 12, 2017
Subjects
math.DG
Other curvature flowsAncient solutionsCurvature 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 Sharp one-sided curvature estimates for mean curvature flow and applications (Mat Langford, 2017)1Langford 2017 Pinched Ancient Solutions to the High Codimension Mean Curvature Flow (Stephen Lynch and Huy The Nguyen, 2017)6Lynch 2017 Convexity estimates for hypersurfaces moving by concave curvature functions (Stephen Lynch, 2020)5Lynch 2020 Collapsing and noncollapsing in convex ancient mean curvature flow (Theodora Bourni, Mat Langford and Stephen Lynch, 2021)8Bourni 2021 Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery (Mat Langford, Stephen Lynch and Huy The Nguyen, 2021)7Langford 2021 Ancient pancake solutions to fully nonlinear curvature flows (Sathya Rengaswami and Mat Langford, 2023)4Rengaswami 2023 Rotational symmetry of ancient solutions to fully nonlinear curvature flows (A. Cogo, S. Lynch and O. Vičánek Martínez, 2023)3Cogo 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. Sharp one-sided curvature estimates for mean curvature flow and applications

    Mat LangfordWider flows

    Topics Curvature pinching, Ancient solutions · Same author Mat Langford · Shared terms “sided curvature”, “sharp estimate”, “curvature estimates”

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

    Mat Langford and James McCoyWider flows

    Topics Curvature pinching, Homogeneous spaces, Other curvature flows · Same author Mat Langford · Shared terms “fully nonlinear”, “nonlinear curvature”, “hypersurfaces evolving”

  3. Rotational symmetry of ancient solutions to fully nonlinear curvature flows

    A. Cogo, S. Lynch and O. Vičánek MartínezWider flows

    Topics Ancient solutions, Other curvature flows · Same author Stephen Lynch · Shared terms “nonlinear curvature”, “fully nonlinear”

  4. Ancient pancake solutions to fully nonlinear curvature flows

    Sathya Rengaswami and Mat LangfordWider flows

    Topics Ancient solutions, Other curvature flows · Same author Mat Langford · Shared terms “nonlinear curvature”, “fully nonlinear”

  5. Convexity estimates for hypersurfaces moving by concave curvature functions

    Stephen LynchWider flows

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

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

    Stephen Lynch and Huy The NguyenWider flows

    Topics Curvature pinching, Ancient solutions · Same author Stephen Lynch · Shared terms “shrinking sphere”

  7. Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery

    Mat Langford, Stephen Lynch and Huy The NguyenWider flows

    Topics Curvature pinching · Same authors Mat Langford, Stephen Lynch

  8. Collapsing and noncollapsing in convex ancient mean curvature flow

    Theodora Bourni, Mat Langford and Stephen LynchWider flows

    Topics Ancient solutions · Same authors Mat Langford, Stephen Lynch · Shared terms “characterisation”

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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.