Paper · Jun 26, 2024

Fattening in mean curvature flow

Tom Ilmanen, Brian White

math.DGarXiv:2406.18703v3Ars Inveniendi Analytica (2025), Paper No. 4, 32 pp
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Abstract

For each , we prove existence of a compact, connected, smoothly embedded, genus- surface with the following property: under mean curvature flow, there is exactly one singular point at the first singular time, and the tangent flow at the singularity is given by a shrinker with genus and with two ends. Furthermore, we show that if is sufficiently large, then fattens at the first singular time. As , the shrinker converges to a multiplicity plane.

Comments: 32 pages, 2 figures. This the version published in Ars Inveniendi Analytica

Mean curvature flowRicci solitonsSingularities

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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 Genus one singularities in mean curvature flow (Adrian Chun-Pong Chu and Ao Sun, 2023)7Chu 2023 Generating Shrinkers by Mean Curvature Flow (David Hoffman, Francisco Martin and Brian White, 2025)2Hoffman 2025 The Genus-Decreasing Property of Mean Curvature Flow, I (Brian White, 2026)4White 2026 Singularities of Mean Curvature Flow of Surfaces (Tom Ilmanen, 2026)1Ilmanen 2026 Lectures on Mean Curvature Flow and Related Equations (Tom Ilmanen, 2026)8Ilmanen 2026 Expanders for Mean Curvature Flow and Counterexamples to Ilmanen's Genus-Reduction Conjecture (David Hoffman, Francisco Martin and Brian White, 2026)3Hoffman 2026 Closed mean curvature flows with prescribed tangent flows (Jingwen Chen, Tang-Kai Lee, Ao Sun et al., 2026)6Chen 2026 Genus Increase in Closed Mean Curvature Flows (Jingwen Chen, 2026)5Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singularities of Mean Curvature Flow of Surfaces

    Tom Ilmanen PDF

    Topics Singularities, Ricci solitons · Same author Tom Ilmanen · Shared terms “smoothly embedded”, “singular time”, “multiplicity”

  2. Generating Shrinkers by Mean Curvature Flow

    David Hoffman, Francisco Martin and Brian White PDF

    Topics Singularities, Ricci solitons · Same author Brian White · Shared terms “smoothly embedded”

  3. Expanders for Mean Curvature Flow and Counterexamples to Ilmanen's Genus-Reduction Conjecture

    David Hoffman, Francisco Martin and Brian White PDF

    Topics Singularities, Ricci solitons · Same author Brian White

  4. The Genus-Decreasing Property of Mean Curvature Flow, I

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    Topics Singularities · Same author Brian White · Shared terms “tangent flow”

  5. Genus Increase in Closed Mean Curvature Flows

    Jingwen Chen PDF

    Topics Singularities, Ricci solitons · Shared terms “singular point”, “tangent flow”

  6. Closed mean curvature flows with prescribed tangent flows

    Jingwen Chen, Tang-Kai Lee, Ao Sun et al. PDF

    Topics Singularities, Ricci solitons · Shared terms “tangent flow”

  7. Genus one singularities in mean curvature flow

    Adrian Chun-Pong Chu and Ao SunWider flows PDF

    Topics Singularities

  8. Lectures on Mean Curvature Flow and Related Equations

    Tom IlmanenWider flows PDF

    Topics Singularities · Same author Tom Ilmanen

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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)

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