Paper · Jan 26, 2026

Ricci Flow on CP1-bundles over a Product of Kähler-Einstein Manifolds

Frederick Tsz-Ho Fong, Hung Tran

math.DGarXiv:2601.18931
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Abstract

In this paper, we study the Ricci flow on CP1-bundles over a product of Kähler-Einstein manifolds whose initial metric is constructed by the ansatz used in works by M. Wang et. al. We prove that the ansatz is preserved along the Ricci flow. Furthermore, in the Kähler case, we proved that Type I finite-time singularity must occur under such an ansatz.

Ricci flowEinstein metricsSingularitiesKähler geometryKähler–Einstein & K-stability

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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 Ricci Flow on Torus Bundles (Dmytro Yeroshkin, 2021)3Yeroshkin 2021 L^∞ estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation (Wangjian Jian and Yalong Shi, 2023)5Jian 2023 Fano Fibrations and Twisted Kähler-Einstein Metrics I (Alexander Bednarek, 2025)6Bednarek 2025 Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow (Alexander Bednarek, 2025)7Bednarek 2025 Singularity Models of Finite-Time Kähler-Ricci Flows (Frederick Tsz-Ho Fong and Hung Tran, 2026)1Fong 2026 Finite-Time Singularities of the Kähler–Ricci Flow on a ℂP^m-Bundle over a Product of Kähler–Einstein Manifolds (Yifan Xiao, 2026)2Xiao 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II (Wangjian Jian and Jian Song, 2026)8Jian 2026 Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class (Frederick Tsz-Ho Fong and Junbang Liu, 2026)4Fong 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Singularity Models of Finite-Time Kähler-Ricci Flows

    Frederick Tsz-Ho Fong and Hung Tran PDF

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Same authors Frederick Tsz-Ho Fong, Hung Tran · Shared terms “finite time”

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

    Yifan Xiao PDF

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

  3. Ricci Flow on Torus Bundles

    Dmytro Yeroshkin PDF

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry

  4. Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class

    Frederick Tsz-Ho Fong and Junbang Liu PDF

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Same author Frederick Tsz-Ho Fong

  5. estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation

    Wangjian Jian and Yalong Shi PDF

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry

  6. Fano Fibrations and Twisted Kähler-Einstein Metrics I

    Alexander Bednarek PDF

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

  7. Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow

    Alexander Bednarek PDF

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

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

    Wangjian Jian and Jian Song PDF

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

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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) 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) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

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    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.

  2. 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.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.