Paper · May 10, 2026

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

David Hoffman, Francisco Martin, Brian White

math.DGarXiv:2605.09657
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Abstract

We construct new expanders for mean curvature flow that are smoothly asymptotic to cones arising from certain shrinkers. For each such cone, we prove the existence of expanders of arbitrarily large genus. Thus, for a fixed incoming shrinker, the genus of the outgoing expander can be chosen much larger than the genus before the singularity, contrary to Ilmanen's genus-reduction conjecture.

Comments: 43 pages, 1 figure

Mean curvature flowRicci solitonsSingularities

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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)6Hoffman 2023 Translating Annuli for Mean Curvature Flow (David Hoffman, Francisco Martín and Brian White, 2023)5Hoffman 2023 Fattening in mean curvature flow (Tom Ilmanen and Brian White, 2024)3Ilmanen 2024 Classification of Semigraphical Translators (Francisco Martín, Mariel Sáez, Raphael Tsiamis et al., 2024)7Martín 2024 Generating Shrinkers by Mean Curvature Flow (David Hoffman, Francisco Martin and Brian White, 2025)2Hoffman 2025 Uniqueness of tangent planes and (non-)removable singularities at infinity for collapsed translators (Eddygledson Souza Gama, Francisco Martín and Niels Martin Møller, 2025)8Gama 2025 The Genus-Decreasing Property of Mean Curvature Flow, I (Brian White, 2026)4White 2026 Genus Increase in Closed Mean Curvature Flows (Jingwen Chen, 2026)1Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Genus Increase in Closed Mean Curvature Flows

    Jingwen Chen PDF

    Topics Singularities, Ricci solitons · Shared terms “genus reduction”, “Ilmanen genus”, “reduction conjecture”

  2. Generating Shrinkers by Mean Curvature Flow

    David Hoffman, Francisco Martin and Brian White PDF

    Topics Singularities, Ricci solitons · Same authors David Hoffman, Francisco Martin, Brian White

  3. Fattening in mean curvature flow

    Tom Ilmanen and Brian White PDF

    Topics Singularities, Ricci solitons · Same author Brian White

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

    Brian WhiteWider flows PDF

    Topics Singularities · Same author Brian White

  5. 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

  6. Annuloids and -wings

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

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

  7. 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

  8. Uniqueness of tangent planes and (non-)removable singularities at infinity for collapsed translators

    Eddygledson Souza Gama, Francisco Martín and Niels Martin MøllerWider flows PDF

    Topics Singularities · Same author Francisco Martin

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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