Paper · Feb 27, 2025

Generating Shrinkers by Mean Curvature Flow

David Hoffman, Francisco Martin, Brian White

math.DGarXiv:2502.20340v3
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Abstract

We prove existence for many examples of shrinkers by producing compact, smoothly embedded surfaces that, under mean curvature flow, develop singularities at which the shrinkers occur as blowups.

Comments: 50 pages, 9 figures. The new version (January 20, 2026) incorporates many small changes suggested by the referees

Mean curvature flowRicci solitonsSingularities

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 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 Annuloids and Δ-wings (D. Hoffman, F. Martin and B. White, 2023)5Hoffman 2023 Translating Annuli for Mean Curvature Flow (David Hoffman, Francisco Martín and Brian White, 2023)4Hoffman 2023 Fattening in mean curvature flow (Tom Ilmanen and Brian White, 2024)2Ilmanen 2024 Classification of Semigraphical Translators (Francisco Martín, Mariel Sáez, Raphael Tsiamis et al., 2024)6Martín 2024 The Genus-Decreasing Property of Mean Curvature Flow, I (Brian White, 2026)7White 2026 Singularities of Mean Curvature Flow of Surfaces (Tom Ilmanen, 2026)3Ilmanen 2026 Expanders for Mean Curvature Flow and Counterexamples to Ilmanen's Genus-Reduction Conjecture (David Hoffman, Francisco Martin and Brian White, 2026)1Hoffman 2026 Closed mean curvature flows with prescribed tangent flows (Jingwen Chen, Tang-Kai Lee, Ao Sun et al., 2026)8Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 authors David Hoffman, Francisco Martin, Brian White

  2. Fattening in mean curvature flow

    Tom Ilmanen and Brian White PDF

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

  3. Singularities of Mean Curvature Flow of Surfaces

    Tom Ilmanen PDF

    Topics Singularities, Ricci solitons · Shared terms “smoothly embedded”

  4. Translating Annuli for Mean Curvature Flow

    David Hoffman, Francisco Martín and Brian WhiteWider flows PDF

    Topics Mean curvature flow · Same authors David Hoffman, Francisco Martin, Brian White

  5. Annuloids and -wings

    D. Hoffman, F. Martin and B. WhiteWider flows PDF

    Topics Mean curvature flow · Same authors David Hoffman, Francisco Martin, Brian White

  6. Classification of Semigraphical Translators

    Francisco Martín, Mariel Sáez, Raphael Tsiamis et al.Wider flows PDF

    Topics Mean curvature flow · Same authors Francisco Martin, Brian White

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

    Brian WhiteWider flows PDF

    Topics Singularities · Same author Brian White

  8. Closed mean curvature flows with prescribed tangent flows

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

    Topics Singularities, Ricci solitons

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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