Paper · Oct 26, 2026

Genus Increase in Closed Mean Curvature Flows

Jingwen Chen

math.DGarXiv:2610.35000v2
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Abstract

We construct closed mean curvature flows with arbitrarily large genus increase in every smooth Riemannian three-manifold, giving counterexamples to Ilmanen's genus-reduction conjecture. We prove that a compact hypersurface with a single isolated conical singularity and a power rate of convergence to its cone admits a mean curvature flow whose forward rescaled limit is a prescribed expander in the tangent space at the singular point, smoothly asymptotic to that cone. Combining this theorem with closed mean curvature flows with prescribed tangent flows (arXiv:2405.15577, arXiv:2607.23879) and the noncompact counterexample (arXiv:2605.09657) gives the desired examples.

Comments: 21 pages. Comments are welcome! v2: Added the Riemannian case

Mean curvature flowRicci solitonsSingularities3-manifoldsBoundaries & non-compact

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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 Ancient solutions to the Ricci flow in dimension 3 (S. Brendle, 2018)8Brendle 2018 Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities (Ao Sun and Jinxin Xue, 2021)7Sun 2021 Uniqueness of conical singularities for mean curvature flows (Tang-Kai Lee and Xinrui Zhao, 2023)6Lee 2023 Mean Curvature Flow from Conical Singularities (Otis Chodosh, J. M. Daniels-Holgate and Felix Schulze, 2023)5Chodosh 2023 Closed mean curvature flows with asymptotically conical singularities (Tang-Kai Lee and Xinrui Zhao, 2024)3Lee 2024 Fattening in mean curvature flow (Tom Ilmanen and Brian White, 2024)4Ilmanen 2024 Expanders for Mean Curvature Flow and Counterexamples to Ilmanen's Genus-Reduction Conjecture (David Hoffman, Francisco Martin and Brian White, 2026)2Hoffman 2026 Closed mean curvature flows with prescribed tangent flows (Jingwen Chen, Tang-Kai Lee, Ao Sun et al., 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. Closed mean curvature flows with prescribed tangent flows

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

    Topics Boundaries & non-compact, Singularities, Ricci solitons · Same author Jingwen Chen · Shared terms “closed mean”, “prescribed tangent”, “tangent flow”

  2. 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 · Shared terms “genus reduction”, “Ilmanen genus”, “reduction conjecture”

  3. Closed mean curvature flows with asymptotically conical singularities

    Tang-Kai Lee and Xinrui Zhao PDF

    Topics Boundaries & non-compact, Singularities, Ricci solitons · Shared terms “closed mean”, “conical singularities”, “Ilmanen”

  4. Fattening in mean curvature flow

    Tom Ilmanen and Brian White PDF

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

  5. Mean Curvature Flow from Conical Singularities

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

    Topics Boundaries & non-compact, Singularities · Shared terms “conical singularities”, “isolated conical”, “Ilmanen”

  6. Uniqueness of conical singularities for mean curvature flows

    Tang-Kai Lee and Xinrui ZhaoWider flows PDF

    Topics Boundaries & non-compact, Singularities · Shared terms “conical singularities”, “tangent flow”

  7. Generic Dynamics of Mean Curvature Flows with Asymptotically Conical Singularities

    Ao Sun and Jinxin Xue PDF

    Topics Boundaries & non-compact, Singularities, Ricci solitons · Shared terms “conical singularities”, “rescaled”

  8. Ancient solutions to the Ricci flow in dimension 3

    S. BrendleThe canon PDF

    Topics 3-manifolds, Boundaries & non-compact, Singularities

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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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  2. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.

  3. 10A.4
    Homotopy Groups, Degree and the Hurewicz TheoremBook 10A · Three-Manifolds and Geometrization

    Spheres mapped into manifolds: the algebraic topology that finite extinction runs on.