Paper · Mar 30, 2021

Local convexity estimates for mean curvature flow

Mat Langford

math.DGWider flowsarXiv:2103.16728
View PDF View HTML arXiv page

As on arXiv

Abstract

We develop a local version of Huisken-Stampacchia iteration, using it to obtain local versions of a host of important sharp curvature pinching estimates for mean curvature flow. The local estimates we obtain do not depend on the quality of noncollapsing of the solution and the method adopted applies in a host of other settings.

View PDF View HTML arXiv page
Submitted
Mar 30, 2021
Subjects
math.DG
Mean curvature flow(Non)collapsingCurvature pinching

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 Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions (Mat Langford and Stephen Lynch, 2017)4Langford 2017 Quadratically pinched hypersurfaces of the sphere via mean curvature flow with surgery (Mat Langford and Huy The Nguyen, 2020)7Langford 2020 Convexity estimates for hypersurfaces moving by concave curvature functions (Stephen Lynch, 2020)5Lynch 2020 Sharp pinching estimates for mean curvature flow in the sphere (Mat Langford and Huy The Nguyen, 2020)3Langford 2020 Collapsing and noncollapsing in convex ancient mean curvature flow (Theodora Bourni, Mat Langford and Stephen Lynch, 2021)8Bourni 2021 Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows (Mat Langford and James McCoy, 2025)2Langford 2025 Local Convexity and Planarity Estimates for Mean Curvature Flow (Liang Cheng, 2026)6Cheng 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 mean curvature flow and applications

    Mat LangfordWider flows

    Topics Curvature pinching · Same author Mat Langford · Shared terms “convexity estimate”, “pinching estimate”, “Huisken”

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

    Mat Langford and James McCoyWider flows

    Topics Curvature pinching · Same author Mat Langford · Shared terms “sharp curvature”, “Stampacchia iteration”, “pinching estimate”

  3. Sharp pinching estimates for mean curvature flow in the sphere

    Mat Langford and Huy The NguyenWider flows

    Topics Curvature pinching · Same author Mat Langford · Shared terms “convexity estimate”, “pinching estimate”

  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 · Same author Mat Langford · Shared terms “Stampacchia iteration”, “pinching estimate”, “Huisken”

  5. Convexity estimates for hypersurfaces moving by concave curvature functions

    Stephen LynchWider flows

    Topics Curvature pinching · Shared terms “convexity estimate”

  6. Local Convexity and Planarity Estimates for Mean Curvature Flow

    Liang ChengWider flows

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

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

    Mat Langford and Huy The NguyenWider flows

    Topics Curvature pinching · Same author Mat Langford · Shared terms “Huisken”

  8. Collapsing and noncollapsing in convex ancient mean curvature flow

    Theodora Bourni, Mat Langford and Stephen LynchWider flows

    Topics (Non)collapsing · Same author Mat Langford · Shared terms “noncollapsing”

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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) 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.