Paper · May 7, 2024

Collapsing immortal Kähler-Ricci flows

Hans-Joachim Hein, Man-Chun Lee, Valentino Tosatti

math.DGarXiv:2405.04208v2Forum Math. Pi 13 (2025), Paper No. e18
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Abstract

We consider the Kähler-Ricci flow on compact Kähler manifolds with semiample canonical bundle and intermediate Kodaira dimension, and show that the flow collapses to a canonical metric on the base of the Iitaka fibration in the locally smooth topology and with bounded Ricci curvature away from the singular fibers. This follows from an asymptotic expansion for the evolving metrics, in the spirit of recent work of the first and third-named authors on collapsing Calabi-Yau metrics, and proves two conjectures of Song and Tian.

Comments: 90 pages; final version to appear in Forum of Math Pi

Ricci flowKähler–Ricci flowRicci-flat & Calabi–Yau(Non)collapsingKähler geometryLong-time existence & convergenceAnalytic minimal model program

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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 On almost nonpositive k-Ricci curvature (Kai Tang, 2021)6Tang 2021 Curvature Estimates for the Continuity Method (Hosea Wondo, 2021)7Wondo 2021 On the Ricci curvature of Kahler-Ricci Flow (Cheuk Yan Fung, 2021)5Fung 2021 Immortal solutions of the Kähler-Ricci flow (Valentino Tosatti, 2024)3Tosatti 2024 Gromov-Hausdorff limits of immortal Kähler-Ricci flows (Man-Chun Lee, Valentino Tosatti and Junsheng Zhang, 2026)1Lee 2026 Higher-order Ricci estimates along immortal Kähler-Ricci flows (Wenrui Kong, 2026)2Kong 2026 Convergence of Scalar Curvature of Long Time Kähler-Ricci Flow on Kähler Manifold (Lei Zhang and Zhenlei Zhang, 2026)4Zhang 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles (Wangjian Jian and Jian Song, 2026)8Jian 2026
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  1. Gromov-Hausdorff limits of immortal Kähler-Ricci flows

    Man-Chun Lee, Valentino Tosatti and Junsheng Zhang PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Long-time existence & convergence · Same authors Man-Chun Lee, Valentino Tosatti · Shared terms “immortal kahler”, “semiample canonical”, “canonical bundle”

  2. Higher-order Ricci estimates along immortal Kähler-Ricci flows

    Wenrui Kong PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Long-time existence & convergence · Shared terms “immortal kahler”, “intermediate Kodaira”, “singular fibers”

  3. Immortal solutions of the Kähler-Ricci flow

    Valentino Tosatti PDF

    Topics Kähler–Ricci flow, Long-time existence & convergence, Kähler geometry · Same author Valentino Tosatti · Shared terms “immortal”, “compact Kähler”

  4. Convergence of Scalar Curvature of Long Time Kähler-Ricci Flow on Kähler Manifold

    Lei Zhang and Zhenlei Zhang PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “semiample canonical”, “Kodaira dimension”, “canonical bundle”

  5. On the Ricci curvature of Kahler-Ricci Flow

    Cheuk Yan Fung PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Long-time existence & convergence · Shared terms “Kodaira dimension”

  6. On almost nonpositive -Ricci curvature

    Kai Tang PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “compact Kähler”

  7. Curvature Estimates for the Continuity Method

    Hosea Wondo PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Long-time existence & convergence · Shared terms “compact Kähler”

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

    Wangjian Jian and Jian Song PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, (Non)collapsing

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-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

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.