Paper · Sep 7, 2026

Finite-Time Singularities of the Kähler–Ricci Flow on a -Bundle over a Product of Kähler–Einstein Manifolds

Yifan Xiao

math.DGarXiv:2609.07124v2
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Abstract

In this paper, we study the Kähler–Ricci flow on -bundles over a product of Kähler–Einstein manifolds, starting from an initial metric with Calabi symmetry. We prove that every finite-time singularity arising along the flow must be of Type I.

Comments: 31 pages, no figures. v2: added related-work discussion on recent results of Fong–Tran and Jian–Song, clarifying that the main conclusion overlaps with theirs while the proof here is independent and uses a different strategy; updated references. No changes to the main results or proofs

Ricci flowKähler–Ricci flowEinstein metricsSingularitiesKähler geometryKähler–Einstein & K-stability

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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 Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities (Alexander Bednarek, 2024)5Bednarek 2024 Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds (Frederick Tsz-Ho Fong and Hung Tran, 2026)1Fong 2026 Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces (Tongxin Xu and Zhenlei Zhang, 2026)8Xu 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles (Wangjian Jian and Jian Song, 2026)7Jian 2026 Singularity Models of Finite-Time Kähler-Ricci Flows (Frederick Tsz-Ho Fong and Hung Tran, 2026)2Fong 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)6Xu 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II (Wangjian Jian and Jian Song, 2026)3Jian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds

    Frederick Tsz-Ho Fong and Hung Tran PDF

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Shared terms “time singularity”, “finite time”, “initial metric”

  2. Singularity Models of Finite-Time Kähler-Ricci Flows

    Frederick Tsz-Ho Fong and Hung Tran PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “finite time”

  3. Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II

    Wangjian Jian and Jian Song PDF

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · 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, Kähler geometry, Singularities · Shared terms “time singularity”, “finite time”

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

    Alexander Bednarek PDF

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

  6. 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, Kähler geometry, Singularities · Shared terms “time singularity”, “finite time”

  7. Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles

    Wangjian Jian and Jian Song PDF

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

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

    Tongxin Xu and Zhenlei Zhang PDF

    Topics Kähler–Ricci flow, 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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.