Paper · Oct 14, 2025

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

Mat Langford, James McCoy

math.DGWider flowsarXiv:2510.12037
PDF Abstract page

As on arXiv

Abstract

In a recent article, a localization of the Huisken–Stampacchia iteration method was developed, and used to establish localizations of the well-known "umbilic", "convexity" and "cylindrical" estimates for hypersurfaces evolving in Euclidean space by mean curvature flow. Here, we adapt the methods developed there to treat more general (fully nonlinear) flows, establishing localizations of asymptotically sharp curvature pinching estimates for hypersurfaces evolving by one-homogeneous functions of curvature under very general conditions. We also briefly describe how the method can be adapted to treat the deformation of hypersurfaces in curved ambient spaces (by suitable speed functions), which is fundamental for many important applications of such flows.

Mean curvature flowOther curvature flowsCurvature 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 2021, 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 Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery (Mat Langford, Stephen Lynch and Huy The Nguyen, 2021)6Langford 2021 Ancient pancake solutions to fully nonlinear curvature flows (Sathya Rengaswami and Mat Langford, 2023)1Rengaswami 2023 Pinching estimates of hypersurfaces by a generalized Gauss curvature flow (Jinrong Hu and Ping Zhang, 2023)3Hu 2023 Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere (Artemis A. Vogiatzi, 2024)5Vogiatzi 2024 Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows (Weimin Sheng and Jiazhuo Yang, 2025)2Sheng 2025 Noncollapsing for Curvature Flows with Inhomogeneous Speeds (Weimin Sheng and Ye Zhu, 2025)8Sheng 2025 The quermassintegral preserving curvature flow for horo-convex hypersurfaces in the sphere (Sara Albert-Niclòs, Esther Cabezas-Rivas and Shujing Pan, 2026)4Albert-Niclò 2026 Local Convexity and Planarity Estimates for Mean Curvature Flow (Liang Cheng, 2026)7Cheng 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ancient pancake solutions to fully nonlinear curvature flows

    Sathya Rengaswami and Mat LangfordWider flows PDF

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

  2. Long time behavior of a class of non-homogeneous anisotropic fully nonlinear curvature flows

    Weimin Sheng and Jiazhuo YangWider flows PDF

    Topics Homogeneous spaces, Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”

  3. Pinching estimates of hypersurfaces by a generalized Gauss curvature flow

    Jinrong Hu and Ping ZhangWider flows PDF

    Topics Curvature pinching, Homogeneous spaces, Other curvature flows · Shared terms “pinching estimate”, “speed function”

  4. The quermassintegral preserving curvature flow for horo-convex hypersurfaces in the sphere

    Sara Albert-Niclòs, Esther Cabezas-Rivas and Shujing PanWider flows PDF

    Topics Curvature pinching, Homogeneous spaces, Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”

  5. Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere

    Artemis A. VogiatziWider flows PDF

    Topics Curvature pinching · Shared terms “cylindrical estimate”, “Stampacchia iteration”

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

    Mat Langford, Stephen Lynch and Huy The NguyenWider flows PDF

    Topics Curvature pinching · Same author Mat Langford

  7. Local Convexity and Planarity Estimates for Mean Curvature Flow

    Liang ChengWider flows PDF

    Topics Curvature pinching · Shared terms “pinching estimate”, “Euclidean”

  8. Noncollapsing for Curvature Flows with Inhomogeneous Speeds

    Weimin Sheng and Ye ZhuWider flows PDF

    Topics Homogeneous spaces, Other curvature flows · Shared terms “nonlinear curvature”, “fully nonlinear”, “speed function”

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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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) 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.