Paper · Apr 14, 2020

High Codimension Mean Curvature Flow with Surgery

Stephen Lynch, Huy The Nguyen

math.DGWider flowsarXiv:2004.07163v3
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Abstract

We construct a mean curvature flow with surgery for submanifolds of arbitrary codimension. The theory applies to closed submanifolds satisfying a natural quadratic pinching condition, which serves as the high-codimension analogue of 2-convexity and is preserved under the flow in dimensions . Our results therefore are in line with the current state-of-the-art in codimension one (where at present 2-convexity is required for surgery). Central to our analysis is a collection of new a priori estimates for the second fundamental form, uniform across surgeries, which yield a precise description of high-curvature regions and permit controlled surgeries. This provides the first notion of mean curvature flow through singularities with topological control in higher codimensions. As a consequence we obtain a sharp classification: Every closed quadratically 2-convexity submanifold is diffeomorphic either to or to a finite connected sum of -bundles over .

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Submitted
Apr 14, 2020
Last revised
Dec 10, 2025 (v3)
Subjects
math.DG

Comments: This replaces an earlier version by the second author and also subsumes the paper arXiv:1805.11808 [math.DG]. It contains a number of new estimates and results

Mean curvature flowSingularitiesSurgeryCurvature pinchingWeak & singular flows

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 Pinched Ancient Solutions to the High Codimension Mean Curvature Flow (Stephen Lynch and Huy The Nguyen, 2017)1Lynch 2017 Cylindrical Estimates for High Codimension Mean Curvature Flow (Huy The Nguyen, 2018)6Nguyen 2018 Evolving Pinched Submanifolds of the Sphere by Mean Curvature Flow (Charles Baker and Huy The Nguyen, 2020)8Baker 2020 Convexity Estimates for High Codimension Mean Curvature Flow (Stephen Lynch and Huy The Nguyen, 2020)4Lynch 2020 Quadratically pinched hypersurfaces of the sphere via mean curvature flow with surgery (Mat Langford and Huy The Nguyen, 2020)5Langford 2020 Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery (Mat Langford, Stephen Lynch and Huy The Nguyen, 2021)2Langford 2021 Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds (Artemis A. Vogiatzi and Huy T. Nguyen, 2023)3Vogiatzi 2023 The planarity estimate in dimensions five and six (Stephen Lynch, 2026)7Lynch 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Pinched Ancient Solutions to the High Codimension Mean Curvature Flow

    Stephen Lynch and Huy The NguyenWider flows

    Topics Curvature pinching · Same authors Stephen Lynch, Huy The Nguyen · Shared terms “high codimension”, “codimension mean”, “submanifolds satisfying”

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

    Mat Langford, Stephen Lynch and Huy The NguyenWider flows

    Topics Surgery, Curvature pinching · Same authors Stephen Lynch, Huy The Nguyen · Shared terms “pinching condition”, “quadratically”, “connected sum”

  3. Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds

    Artemis A. Vogiatzi and Huy T. NguyenWider flows

    Topics Curvature pinching, Singularities · Same author Huy The Nguyen · Shared terms “high codimension”, “codimension mean”, “quadratic pinching”

  4. Convexity Estimates for High Codimension Mean Curvature Flow

    Stephen Lynch and Huy The NguyenWider flows

    Topics Mean curvature flow · Same authors Stephen Lynch, Huy The Nguyen · Shared terms “high codimension”, “codimension mean”, “quadratically”

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

    Mat Langford and Huy The NguyenWider flows

    Topics Surgery, Curvature pinching · Same author Huy The Nguyen · Shared terms “pinching condition”, “quadratically”, “connected sum”

  6. Cylindrical Estimates for High Codimension Mean Curvature Flow

    Huy The NguyenWider flows

    Topics Singularities · Same author Huy The Nguyen · Shared terms “high codimension”, “codimension mean”, “quadratic”

  7. The planarity estimate in dimensions five and six

    Stephen LynchWider flows

    Topics Curvature pinching, Singularities · Same author Stephen Lynch · Shared terms “high codimension”, “curvature regions”, “high curvature”

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

    Charles Baker and Huy The NguyenWider flows

    Topics Curvature pinching · Same author Huy The Nguyen · Shared terms “high codimension”, “codimension mean”, “pinching condition”

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) 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) Surgery Stop the flow just before a neck pinches, cut along the thin neck, glue standard caps onto the two ends, and restart. Start with: Four-manifolds with positive isotropic curvature (Hamilton, 1997) 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.10
    Surgery on a δ-NeckBook 12B · κ-Solutions and Surgery

    The gluing construction in detail: the profile function, preservation of the Hamilton–Ivey pinching, closeness to the standard solution, and the definition of Ricci flow with surgery as a space-time (Morgan–Tian chapters 13–15).

  2. 12B.5
    The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery

    Every high-curvature point looks like a neck or a cap.

  3. 12B.7
    The Singular Time: Horns and Strong NecksBook 12B · κ-Solutions and Surgery

    The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.