Paper · May 30, 2018

Cylindrical Estimates for High Codimension Mean Curvature Flow

Huy The Nguyen

math.DGWider flowsarXiv:1805.11808
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Abstract

We study high codimension mean curvature flow of a submanifold of dimension in Euclidean space subject to the quadratic curvature condition . This condition extends the notion of two-convexity for hypersurfaces to high codimension submanifolds. We analyse singularity formation in the mean curvature flow of high codimension by directly proving a pointwise gradient estimate. We then show that near a singularity the surface is quantitatively cylindrical.

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Submitted
May 30, 2018
Subjects
math.DG, math.AP
Mean curvature flowSingularities

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)4Lynch 2017 High Codimension Mean Curvature Flow with Surgery (Stephen Lynch and Huy The Nguyen, 2020)2Lynch 2020 Evolving Pinched Submanifolds of the Sphere by Mean Curvature Flow (Charles Baker and Huy The Nguyen, 2020)6Baker 2020 Convexity Estimates for High Codimension Mean Curvature Flow (Stephen Lynch and Huy The Nguyen, 2020)3Lynch 2020 Singularity Models for High Codimension Mean Curvature Flow in Riemannian Manifolds (Artemis A. Vogiatzi and Huy T. Nguyen, 2023)1Vogiatzi 2023 High Codimension Mean Curvature Flow in the Complex Projective Space (Artemis A. Vogiatzi, 2023)5Vogiatzi 2023 High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension (Ben Andrews and Qiyu Zhou, 2026)7Andrews 2026 High Codimension Curve Shortening Flow with Free Boundary (Huy The Nguyen and Artemis A. Vogiatzi, 2026)8Nguyen 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

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

  2. High Codimension Mean Curvature Flow with Surgery

    Stephen Lynch and Huy The NguyenWider flows

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

  3. Convexity Estimates for High Codimension Mean Curvature Flow

    Stephen Lynch and Huy The NguyenWider flows

    Topics Mean curvature flow · Same author Huy The Nguyen · Shared terms “high codimension”, “codimension mean”

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

    Stephen Lynch and Huy The NguyenWider flows

    Topics Mean curvature flow · Same author Huy The Nguyen · Shared terms “high codimension”, “codimension mean”

  5. High Codimension Mean Curvature Flow in the Complex Projective Space

    Artemis A. VogiatziWider flows

    Topics Singularities · Shared terms “high codimension”, “codimension mean”, “quadratic”

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

    Charles Baker and Huy The NguyenWider flows

    Topics Mean curvature flow · Same author Huy The Nguyen · Shared terms “high codimension”, “codimension mean”, “curvature condition”

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

    Ben Andrews and Qiyu ZhouWider flows

    Topics Singularities · Shared terms “high codimension”, “codimension mean”, “Euclidean space”

  8. High Codimension Curve Shortening Flow with Free Boundary

    Huy The Nguyen and Artemis A. VogiatziWider flows

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

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)

In the guidebook

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    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  2. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.

  3. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.