Paper · Oct 6, 2026

Increase in topological complexity along the mean curvature flow

Natasha Diederen

math.DGWider flowsarXiv:2610.08357
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Abstract

We construct an example of a smooth embedding of in whose evolution under the mean curvature flow forms an isolated singularity at one point, after which it is a smoothly embedded copy of . In doing so, we prove some general results about -invariant mean curvature flow whose initial data is generated by the rotation of a graphical profile curve, including an analysis of bubblesheet singularities. The most involved part of our analysis is an instant smoothness result, which requires a delicate approximation and pseudolocality.

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Submitted
Oct 6, 2026
Subjects
math.DG, math.AP
Mean curvature flowSingularities

Its neighbourhood

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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 Type II smoothing in mean curvature flow (Sigurd Angenent, Panagiota Daskalopoulos and Natasa Sesum, 2021)1Angenent 2021 Mean curvature flow with generic low-entropy initial data II (Otis Chodosh, Christos Mantoulidis and Felix Schulze, 2023)6Chodosh 2023 Mean Curvature Flow from Conical Singularities (Otis Chodosh, J. M. Daniels-Holgate and Felix Schulze, 2023)8Chodosh 2023 Generating Shrinkers by Mean Curvature Flow (David Hoffman, Francisco Martin and Brian White, 2025)3Hoffman 2025 Singularities of Mean Curvature Flow of Surfaces (Tom Ilmanen, 2026)4Ilmanen 2026 Compact Mean Curvature Flows from Non-Compact Initial Data (Theodora Bourni, Nathan Burns and Marcus Flook, 2026)5Bourni 2026 Singularities of Mean Curvature Flow with Bounded Total Curvature (Yongheng Han and Zhen Wang, 2026)7Han 2026 Genus Increase in Closed Mean Curvature Flows (Jingwen Chen, 2026)2Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Type II smoothing in mean curvature flow

    Sigurd Angenent, Panagiota Daskalopoulos and Natasa SesumWider flows

    Topics Singularities · Shared terms “invariant Mean”, “isolated singularity”

  2. Genus Increase in Closed Mean Curvature Flows

    Jingwen Chen

    Topics Singularities · Shared terms “increase”, “isolated”

  3. Generating Shrinkers by Mean Curvature Flow

    David Hoffman, Francisco Martin and Brian White

    Topics Singularities · Shared terms “smoothly embedded”

  4. Singularities of Mean Curvature Flow of Surfaces

    Tom Ilmanen

    Topics Singularities · Shared terms “smoothly embedded”

  5. Compact Mean Curvature Flows from Non-Compact Initial Data

    Theodora Bourni, Nathan Burns and Marcus FlookWider flows

    Topics Singularities · Shared terms “initial data”

  6. Mean curvature flow with generic low-entropy initial data II

    Otis Chodosh, Christos Mantoulidis and Felix SchulzeWider flows

    Topics Singularities · Shared terms “initial data”

  7. Singularities of Mean Curvature Flow with Bounded Total Curvature

    Yongheng Han and Zhen WangWider flows

    Topics Singularities · Shared terms “smoothness”

  8. Mean Curvature Flow from Conical Singularities

    Otis Chodosh, J. M. Daniels-Holgate and Felix SchulzeWider flows

    Topics Singularities · Shared terms “isolated”, “smoothness”

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)

In the guidebook

  1. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    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.