Paper · Feb 23, 2026

High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension

Ben Andrews, Qiyu Zhou

math.DGWider flowsarXiv:2602.19537v2
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Abstract

In the pseudo-Euclidean space , we consider the mean curvature flow of -dimensional spacelike submanifolds with spacelike codimension one and arbitrary timelike codimension . We show that if the initial submanifold is compact and spacelike-convex (the acceleration along every geodesic is strictly spacelike), then natural quantities measuring curvature pinching and noncollapsing are preserved under the flow. Moreover, we prove an analogue of the Huisken and Gage-Hamilton theorems in this setting, which states that the mean curvature flow deforms any such submanifold to a point in finite time, and that the solution is asymptotic to a shrinking sphere in a maximally spacelike affine subspace .

Mean curvature flowSingularities(Non)collapsingCurvature pinchingPhysics

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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 Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds (Artemis A. Vogiatzi and Huy T. Nguyen, 2023)1Vogiatzi 2023 Collapsing of Mean Curvature Flow of Hypersurfaces to Complex Submanifolds (Farnaz Ghanbari and Samreena, 2023)7Ghanbari 2023 Mean Curvature Flow of High Codimension in Complex Projective Space (Artemis A. Vogiatzi, 2023)2Vogiatzi 2023 Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere (Artemis A. Vogiatzi, 2024)4Vogiatzi 2024 A mean curvature flow approach to Hamilton's pinching theorem (Liang Cheng and Zhenyu Lu, 2026)8Cheng 2026 Spacelike Brakke flows with boundary in pseudo-Euclidean space (Javier Hervás, 2026)6Hervás 2026 The planarity estimate in dimensions five and six (Stephen Lynch, 2026)5Lynch 2026 Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow (Tsz-Kiu Aaron Chow, 2026)3Chow 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds

    Artemis A. Vogiatzi and Huy T. NguyenWider flows PDF

    Topics Curvature pinching, Singularities · Shared terms “codimension mean”, “high codimension”, “submanifolds”

  2. Mean Curvature Flow of High Codimension in Complex Projective Space

    Artemis A. VogiatziWider flows PDF

    Topics Curvature pinching, Singularities · Shared terms “high codimension”, “submanifolds”, “Euclidean space”

  3. Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow

    Tsz-Kiu Aaron Chow PDF

    Topics Curvature pinching · Shared terms “codimension mean”, “Euclidean space”

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

    Artemis A. VogiatziWider flows PDF

    Topics Curvature pinching, Singularities · Shared terms “submanifolds”

  5. The planarity estimate in dimensions five and six

    Stephen LynchWider flows PDF

    Topics Curvature pinching, Singularities · Shared terms “high codimension”

  6. Spacelike Brakke flows with boundary in pseudo-Euclidean space

    Javier HervásWider flows PDF

    Topics Physics, Singularities · Shared terms “spacelike”, “pseudo Euclidean”, “Euclidean space”

  7. Collapsing of Mean Curvature Flow of Hypersurfaces to Complex Submanifolds

    Farnaz Ghanbari and SamreenaWider flows PDF

    Topics (Non)collapsing, Singularities · Shared terms “submanifolds”, “finite time”

  8. A mean curvature flow approach to Hamilton's pinching theorem

    Liang Cheng and Zhenyu LuWider flows PDF

    Topics Curvature pinching · Shared terms “Hamilton”

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) (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. 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. 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.

  3. 11A.7
    Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles

    Scalar and tensor maximum principles, and preserved curvature conditions.