Paper · Feb 27, 2025

Non-collapsed finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I

Ronan J. Conlon, Max Hallgren, Zilu Ma

math.DGarXiv:2502.19804v2
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Abstract

We show that any non-collapsed finite time singularity of the Ricci flow on a compact Kähler surface is of Type I. Combined with a previous result of the first author, Cifarelli, and Deruelle, it follows that any such singularity is modeled on the shrinking Ricci soliton of Feldman-Ilmanen-Knopf on the total space of the line bundle .

Comments: 46 pages; Theorem 4.4 added

Ricci flowRicci solitonsSingularities(Non)collapsingKähler geometry

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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 On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces (Charles Cifarelli, Ronan J. Conlon and Alix Deruelle, 2022)3Cifarelli 2022 On noncollapsed F-limit metric solitons (Pak-Yeung Chan, Zilu Ma and Yongjia Zhang, 2024)8Chan 2024 An ε-Regularity Theorem for Non-collapsed Ricci Flow (Harry Fluck and Max Hallgren, 2025)5Fluck 2025 Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program (Longteng Chen, Max Hallgren and Lucas Lavoyer, 2026)7Chen 2026 Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces (Tongxin Xu and Zhenlei Zhang, 2026)6Xu 2026 Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I (Charles Cifarelli, Ronan Conlon, Max Hallgren et al., 2026)1Cifarelli 2026 Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I (Tongxin Xu and Zhenlei Zhang, 2026)2Xu 2026 Finite Time Singularities of the Kähler Ricci Flow on Compact Kähler Surfaces are of Type I (Yeyun Xu and Linfeng Zhou, 2026)4Xu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I

    Charles Cifarelli, Ronan Conlon, Max Hallgren et al. PDF

    Topics (Non)collapsing, Kähler geometry, Singularities · Same authors Ronan J. Conlon, Max Hallgren · Shared terms “Kähler surfaces”, “time singularity”, “compact Kähler”

  2. Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I

    Tongxin Xu and Zhenlei Zhang PDF

    Topics (Non)collapsing, Kähler geometry, Singularities · Shared terms “collapsed finite”, “Kähler surfaces”, “time singularity”

  3. On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces

    Charles Cifarelli, Ronan J. Conlon and Alix Deruelle PDF

    Topics Kähler geometry, Singularities, Ricci solitons · Same author Ronan J. Conlon · Shared terms “Kähler surfaces”, “compact Kähler”, “finite time”

  4. Finite Time Singularities of the Kähler Ricci Flow on Compact Kähler Surfaces are of Type I

    Yeyun Xu and Linfeng Zhou PDF

    Topics (Non)collapsing, Kähler geometry, Singularities · Shared terms “Kähler surfaces”, “compact Kähler”, “time singularity”

  5. An -Regularity Theorem for Non-collapsed Ricci Flow

    Harry Fluck and Max Hallgren PDF

    Topics (Non)collapsing, Kähler geometry, Singularities · Same author Max Hallgren · Shared terms “collapsed”

  6. Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces

    Tongxin Xu and Zhenlei Zhang PDF

    Topics (Non)collapsing, Kähler geometry, Singularities · Shared terms “time singularity”, “finite time”

  7. Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program

    Longteng Chen, Max Hallgren and Lucas Lavoyer PDF

    Topics (Non)collapsing, Kähler geometry, Singularities · Same author Max Hallgren · Shared terms “time singularity”, “finite time”

  8. On noncollapsed -limit metric solitons

    Pak-Yeung Chan, Zilu Ma and Yongjia Zhang PDF

    Topics (Non)collapsing, Singularities, Ricci solitons · Same author Zilu Ma

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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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) 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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

In the guidebook

  1. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.