Paper · Oct 1, 2026

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

Yeyun Xu, Linfeng Zhou

math.DGarXiv:2610.01643
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Abstract

We establish a Type I curvature bound for finite-time volume-collapsing Kähler–Ricci flows on compact Kähler surfaces. No symmetry assumption is imposed, and the initial Kähler class need not be rational. The proof combines local symplectic topology with the classification of gradient Kähler–Ricci shrinking solitons. Bamler's compactness and structure theory provides the shrinking limits used in the argument.

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Submitted
Oct 1, 2026
Subjects
math.DG

Comments: 25 pages

Ricci flowKähler–Ricci flowRicci solitonsSingularities(Non)collapsingKähler geometryLimit spaces & RCD

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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)6Cifarelli 2022 Finite time singularities of the Kähler-Ricci flow (Wangjian Jian, Jian Song and Gang Tian, 2023)7Jian 2023 Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities (Alexander Bednarek, 2024)8Bednarek 2024 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)3Conlon 2025 On the tangent flow to the collapsing Kähler-Ricci flow on Hirzebruch surfaces (Jiangtao Li, 2025)5Li 2025 Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces (Tongxin Xu and Zhenlei Zhang, 2026)4Xu 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
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.

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

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

    Tongxin Xu and Zhenlei Zhang

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

  3. 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

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

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

    Tongxin Xu and Zhenlei Zhang

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

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

    Jiangtao Li

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

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

    Charles Cifarelli, Ronan J. Conlon and Alix Deruelle

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

  7. Finite time singularities of the Kähler-Ricci flow

    Wangjian Jian, Jian Song and Gang Tian

    Topics Kähler–Ricci flow, (Non)collapsing, Limit spaces & RCD · Shared terms “time singularity”, “finite time”

  8. Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities

    Alexander Bednarek

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

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) 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)

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.