Paper · Jan 29, 2026

Singularities of Mean Curvature Flow of Surfaces

Tom Ilmanen

math.DGarXiv:2601.21133
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Abstract

This paper proves that, at the first singular time for a smoothly immersed surface moving by mean curvature flow in a n-manifold, each tangent flow is given by a smooth, branched shrinker, possibly with multiplicity. If n=3 and if the initial surface is embedded, then the shrinker is smoothly embedded without branch points, but possibly with multiplicity. A key ingredient of the proof is a new, local version of the Gauss-Bonnet formula.

Comments: 37 pages. This unpublished paper, written by Tom Ilmanen (1961-2025) in 1993 or 1994, has been posted on arXiv with the permission of his family.

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 On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces (Richard H Bamler and Bruce Kleiner, 2023)6Bamler 2023 Mean curvature flow with multiplicity 2 convergence in ℝ³ (Jingwen Chen and Ao Sun, 2023)8Chen 2023 Fattening in mean curvature flow (Tom Ilmanen and Brian White, 2024)1Ilmanen 2024 Generating Shrinkers by Mean Curvature Flow (David Hoffman, Francisco Martin and Brian White, 2025)2Hoffman 2025 Lectures on Mean Curvature Flow and Related Equations (Tom Ilmanen, 2026)4Ilmanen 2026 Rigidity and gap theorems for Ricci shrinkers (Pak-Yeung Chan and Yongjia Zhang, 2026)7Chan 2026 Closed mean curvature flows with prescribed tangent flows (Jingwen Chen, Tang-Kai Lee, Ao Sun et al., 2026)3Chen 2026 Singularities of Mean Curvature Flow with Bounded Total Curvature (Yongheng Han and Zhen Wang, 2026)5Han 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Fattening in mean curvature flow

    Tom Ilmanen and Brian White 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 · Shared terms “smoothly embedded”

  3. 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”

  4. Lectures on Mean Curvature Flow and Related Equations

    Tom IlmanenWider flows PDF

    Topics Singularities · Same author Tom Ilmanen

  5. Singularities of Mean Curvature Flow with Bounded Total Curvature

    Yongheng Han and Zhen WangWider flows PDF

    Topics Singularities · Shared terms “tangent flow”, “singular time”

  6. On the Multiplicity One Conjecture for Mean Curvature Flows of surfaces

    Richard H Bamler and Bruce KleinerWider flows PDF

    Topics Singularities · Shared terms “multiplicity”

  7. Rigidity and gap theorems for Ricci shrinkers

    Pak-Yeung Chan and Yongjia Zhang PDF

    Topics Singularities, Ricci solitons · Shared terms “local version”

  8. Mean curvature flow with multiplicity 2 convergence in

    Jingwen Chen and Ao SunWider flows PDF

    Topics Mean curvature flow · Shared terms “multiplicity”

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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