Paper · Sep 1, 2020

Sharp pinching estimates for mean curvature flow in the sphere

Mat Langford, Huy The Nguyen

math.DGWider flowsarXiv:2009.00986
View PDF View HTML arXiv page

As on arXiv

Abstract

We prove a suite of asymptotically sharp quadratic curvature pinching estimates for mean curvature flow in the sphere which generalize Simons' rigidity theorem for minimal hypersurfaces. We then obtain derivative estimates for the second fundamental form which we utilize, via a compactness argument, to obtain a convexity estimate. Together, the convexity and cylindrical estimates yield a partial classification of singularity models. We also obtain new rigidity results for ancient solutions.

View PDF View HTML arXiv page
Submitted
Sep 1, 2020
Subjects
math.DG

Comments: arXiv admin note: text overlap with arXiv:2006.08049

Mean curvature flowAncient solutionsSingularitiesCurvature pinchingLimit spaces & RCDStability & uniquenessMinimal & CMC surfaces

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)2Langford 2017 Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions (Mat Langford and Stephen Lynch, 2017)8Langford 2017 Pinched Ancient Solutions to the High Codimension Mean Curvature Flow (Stephen Lynch and Huy The Nguyen, 2017)7Lynch 2017 Evolving Pinched Submanifolds of the Sphere by Mean Curvature Flow (Charles Baker and Huy The Nguyen, 2020)4Baker 2020 Quadratically pinched hypersurfaces of the sphere via mean curvature flow with surgery (Mat Langford and Huy The Nguyen, 2020)1Langford 2020 Local convexity estimates for mean curvature flow (Mat Langford, 2021)5Langford 2021 Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery (Mat Langford, Stephen Lynch and Huy The Nguyen, 2021)3Langford 2021 Local umbilic, convexity and cylindrical estimates for fully nonlinear curvature flows (Mat Langford and James McCoy, 2025)6Langford 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Quadratically pinched hypersurfaces of the sphere via mean curvature flow with surgery

    Mat Langford and Huy The NguyenWider flows

    Topics Curvature pinching, Minimal & CMC surfaces · Same authors Mat Langford, Huy The Nguyen · Shared terms “cylindrical estimate”, “derivative estimates”, “quadratic curvature”

  2. 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 “pinching estimate”, “sharp pinching”, “convexity estimate”

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

    Mat Langford, Stephen Lynch and Huy The NguyenWider flows

    Topics Curvature pinching, Minimal & CMC surfaces · Same authors Mat Langford, Huy The Nguyen · Shared terms “quadratic curvature”

  4. Evolving Pinched Submanifolds of the Sphere by Mean Curvature Flow

    Charles Baker and Huy The NguyenWider flows

    Topics Curvature pinching, Minimal & CMC surfaces · Same author Huy The Nguyen · Shared terms “fundamental form”

  5. Local convexity estimates for mean curvature flow

    Mat LangfordWider flows

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

  6. 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 “cylindrical estimate”, “pinching estimate”, “asymptotically sharp”

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

    Stephen Lynch and Huy The NguyenWider flows

    Topics Curvature pinching, Ancient solutions · Same author Huy The Nguyen · Shared terms “fundamental form”

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

    Mat Langford and Stephen LynchWider flows

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

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) 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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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. 12B.6
    Bounded Curvature at Bounded DistanceBook 12B · κ-Solutions and Surgery

    The heart of the canonical neighbourhood proof: why curvature at bounded rescaled distance from a point of high curvature stays bounded, via incomplete limits, cones and Hamilton–Ivey pinching (Morgan–Tian, chapter 10).

  2. 12B.3
    The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery

    Necks and caps, compactness, and universal estimates.

  3. 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.