Paper · Apr 8, 2025

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

Jiangtao Li

math.DGarXiv:2504.06471
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Abstract

In this paper, we study the collpasing Kähler-Ricci flow on Hirzebruch surfaces, which develops finite time singularities. We show that any tangent flow based at a point in the singular time slice is the Kähler-Ricci flow associated with a nonflat gradient Kähler-Ricci shrinker with finitely many orbifold singularities .

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Submitted
Apr 8, 2025
Subjects
math.DG

Comments: 10 pages

Ricci flowKähler–Ricci flowRicci solitonsSingularities(Non)collapsingKähler geometry

Its neighbourhood

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 Calabi Symmetry and the Continuity Method (Hosea Wondo, 2022)5Wondo 2022 The modified Kähler-Ricci flow, II (Haotian Wu and Zhou Zhang, 2022)8Wu 2022 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)6Hallgren 2023 Finite Time Singularities of Collapsing Kähler Ricci Flow on Ruled Surfaces (Tongxin Xu and Zhenlei Zhang, 2026)1Xu 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles (Wangjian Jian and Jian Song, 2026)7Jian 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 Collapsed Finite Time Singularities of the Kähler-Ricci Flow on Complex Surfaces are of Type I (Tongxin Xu and Zhenlei Zhang, 2026)2Xu 2026 Finite Time Singularities of the Kähler Ricci Flow on Compact Kähler Surfaces are of Type I (Yeyun Xu and Linfeng Zhou, 2026)4Xu 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

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

  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 “time singularity”, “finite time”

  3. 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 “time singularity”, “tangent flow”, “finite time”

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

    Yeyun Xu and Linfeng Zhou

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

  5. Calabi Symmetry and the Continuity Method

    Hosea Wondo

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “Hirzebruch surface”

  6. Geometric regularity of blow-up limits of the Kähler-Ricci flow

    Max Hallgren, Wangjian Jian, Jian Song et al.

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “flow based”, “time slice”

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

    Wangjian Jian and Jian Song

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

  8. The modified Kähler-Ricci flow, II

    Haotian Wu and Zhou Zhang

    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) 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.