Paper · Sep 16, 2026

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

Tongxin Xu, Zhenlei Zhang

math.DGarXiv:2609.18834v2
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Abstract

We prove that any finite-time collapsing Kähler-Ricci flow on compact Kähler surfaces develops a Type I singularity. Together with the previous results, this implies that any finite time singularity of the Kähler-Ricci flow on compact Kähler surfaces is of Type I.

Comments: Added new results on the optimal fiber collapse rate. Comments are welcome!

Ricci flowKähler–Ricci flowSingularities(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)8Cifarelli 2022 The modified Kähler-Ricci flow, II (Haotian Wu and Zhou Zhang, 2022)7Wu 2022 Non-collapsed finite time singularities of the Ricci flow on compact Kähler surfaces are of Type I (Ronan J. Conlon, Max Hallgren and Zilu Ma, 2025)2Conlon 2025 On the tangent flow to the collapsing Kähler-Ricci flow on Hirzebruch surfaces (Jiangtao Li, 2025)6Li 2025 Finite Time Type I Singularities of the Kähler Ricci Flow (Tongxin Xu and Zhenlei Zhang, 2026)4Xu 2026 Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces (Tongxin Xu and Zhenlei Zhang, 2026)1Xu 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)3Cifarelli 2026 Finite Time Singularities of the Kähler Ricci Flow on Compact Kähler Surfaces are of Type I (Yeyun Xu and Linfeng Zhou, 2026)5Xu 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 Collapsing Kähler Ricci Flow on Ruled Surfaces

    Tongxin Xu and Zhenlei Zhang PDF

    Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Same authors Tongxin Xu, Zhenlei Zhang · Shared terms “time singularity”, “finite time”, “surfaces develops”

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

    Ronan J. Conlon, Max Hallgren and Zilu Ma PDF

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

  3. 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 Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Shared terms “Kähler surfaces”, “time singularity”, “compact Kähler”

  4. Finite Time Type I Singularities of the Kähler Ricci Flow

    Tongxin Xu and Zhenlei Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same authors Tongxin Xu, Zhenlei Zhang · Shared terms “finite time”, “compact Kähler”

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

    Yeyun Xu and Linfeng Zhou PDF

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

  6. On the tangent flow to the collapsing Kähler-Ricci flow on Hirzebruch surfaces

    Jiangtao Li PDF

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

  7. The modified Kähler-Ricci flow, II

    Haotian Wu and Zhou Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “time singularity”, “finite time”

  8. 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–Ricci flow, Kähler geometry, Singularities · Shared terms “Kähler surfaces”, “compact Kähler”, “finite time”

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.

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) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) 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. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  2. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

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

  3. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.