Paper · Sep 1, 2026

Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces

Tongxin Xu, Zhenlei Zhang

math.DGarXiv:2609.01442v2
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Abstract

We prove that any finite time collapsing Kähler Ricci flow on ruled surfaces develops a Type I singularity, such singularity is modeled on the standard product shrinker . As an application, we obtain the optimal collapse rate of fibers on ruled surfaces.

Comments: 15 pages

Ricci flowKähler–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 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)8Conlon 2025 On the tangent flow to the collapsing Kähler-Ricci flow on Hirzebruch surfaces (Jiangtao Li, 2025)3Li 2025 Compactness and Rigidity of Complete Kähler Ricci Shrinkers (Tongxin Xu and Zhenlei Zhang, 2026)7Xu 2026 Finite Time Type I Singularities of the Kähler Ricci Flow (Tongxin Xu and Zhenlei Zhang, 2026)2Xu 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)4Cifarelli 2026 Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I (Tongxin Xu and Zhenlei Zhang, 2026)1Xu 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 Kähler-Ricci Shrinkers with Nonnegative Ricci Curvature (Tongxin Xu and Zhenlei Zhang, 2026)6Xu 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I

    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. 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”

  3. 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”

  4. 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 “time singularity”, “finite time”

  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 “time singularity”, “finite time”

  6. Kähler-Ricci Shrinkers with Nonnegative Ricci Curvature

    Tongxin Xu and Zhenlei Zhang PDF

    Topics Kähler geometry, Ricci solitons · Same authors Tongxin Xu, Zhenlei Zhang

  7. Compactness and Rigidity of Complete Kähler Ricci Shrinkers

    Tongxin Xu and Zhenlei Zhang PDF

    Topics Kähler geometry, Ricci solitons · Same authors Tongxin Xu, Zhenlei Zhang

  8. 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 “time singularity”, “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) 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.