Paper · May 7, 2024

Immortal solutions of the Kähler-Ricci flow

Valentino Tosatti

math.DGarXiv:2405.04444v2
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As on arXiv

Abstract

We survey some recent developments on solutions of the Kähler-Ricci flow on compact Kähler manifolds which exist for all positive times.

Comments: 18 pages; submitted to the AMS Contemporary Mathematics volume in memory of Steve Zelditch; final version to appear in Contemp. Math

Ricci flowKähler–Ricci flowKähler geometryLong-time existence & convergence

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 The modified Kähler-Ricci flow, II (Haotian Wu and Zhou Zhang, 2022)5Wu 2022 Collapsing immortal Kähler-Ricci flows (Hans-Joachim Hein, Man-Chun Lee and Valentino Tosatti, 2024)1Hein 2024 Weak transcendental base-point freeness and diameter lower bounds for the Kähler-Ricci flow (Junsheng Zhang, 2025)6Zhang 2025 Gromov-Hausdorff limits of immortal Kähler-Ricci flows (Man-Chun Lee, Valentino Tosatti and Junsheng Zhang, 2026)2Lee 2026 Higher-order Ricci estimates along immortal Kähler-Ricci flows (Wenrui Kong, 2026)4Kong 2026 Asymptotic Profiles and Non-Trivial Breathers in Kahler-Ricci Flow (Longteng Chen, 2026)8Chen 2026 Kähler-Ricci Flow: from divisors to cusps (Eleonora Di Nezza, Vincent Guedj and Chinh H. Lu, 2026)7Di Nezza 2026 Geometric smoothing by the Kähler-Ricci Flow (Eleonora Di Nezza, Vincent Guedj and Chinh Hoang Lu, 2026)3Di Nezza 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Collapsing immortal Kähler-Ricci flows

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

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

  2. Gromov-Hausdorff limits of immortal Kähler-Ricci flows

    Man-Chun Lee, Valentino Tosatti and Junsheng Zhang PDF

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

  3. Geometric smoothing by the Kähler-Ricci Flow

    Eleonora Di Nezza, Vincent Guedj and Chinh Hoang Lu PDF

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “compact Kähler”

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

    Wenrui Kong PDF

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

  5. The modified Kähler-Ricci flow, II

    Haotian Wu and Zhou Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry

  6. Weak transcendental base-point freeness and diameter lower bounds for the Kähler-Ricci flow

    Junsheng Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “compact Kähler”

  7. Kähler-Ricci Flow: from divisors to cusps

    Eleonora Di Nezza, Vincent Guedj and Chinh H. Lu PDF

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “compact Kähler”

  8. Asymptotic Profiles and Non-Trivial Breathers in Kahler-Ricci Flow

    Longteng Chen PDF

    Topics Kähler–Ricci flow, Long-time existence & convergence, Kähler geometry

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) 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.