Paper · Apr 24, 2016

Convergence of weak Kähler-Ricci Flows on minimal models of positive Kodaira dimension

P. Eyssidieux, V. Guedj, A. Zeriahi

math.CVarXiv:1604.07001v2
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Abstract

Studying the behavior of the Kähler-Ricci flow on mildly singular varieties, one is naturally lead to study weak solutions of degenerate parabolic complex Monge-Ampère equations. In this article, the third of a series on this subject, we study the long term behavior of the normalized Kähler-Ricci flow on mildly singular varieties of positive Kodaira dimension, generalizing results of Song and Tian who dealt with smooth minimal models.

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Submitted
Apr 24, 2016
Last revised
Mar 4, 2017 (v2)
Subjects
math.CV, math.DG

Comments: The proof of the parabolic comparison principle in the previous version was incorrect. We provide here a different approach which yields an alternative proof of the global comparison principle in the ample locus

Ricci flowKähler–Ricci flowKähler geometryWeak & singular flowsAnalytic minimal model program

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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 2016, 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 Convergence of scalar curvature of Kahler-Ricci flow on manifolds of positive Kodaira dimension (Wangjian Jian, 2018)3Jian 2018 Pluripotential kahler-ricci flows (Vincent Guedj, Hoang Chinh Lu and Ahmed Zeriahi, 2018)1Guedj 2018 The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampere flows (Vincent Guedj, Hoang Chinh Lu and Ahmed Zeriahi, 2018)2Guedj 2018 Stability of solutions to complex Monge-Ampere flows (Vincent Guedj, Hoang Chinh Lu and Ahmed Zeriahi, 2018)5Guedj 2018 Convergence of the weak Kähler-Ricci Flow on manifolds of general type (Tat Dat Tô, 2019)6Tô 2019 On the Ricci curvature of Kahler-Ricci Flow (Cheuk Yan Fung, 2021)8Fung 2021 Kähler-Ricci flows coming out of metric spaces (Alix Deruelle, Vincent Guedj, Henri Guenancia et al., 2025)4Deruelle 2025 Convergence of Scalar Curvature of Long Time Kähler-Ricci Flow on Kähler Manifold (Lei Zhang and Zhenlei Zhang, 2026)7Zhang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Pluripotential kahler-ricci flows

    Vincent Guedj, Hoang Chinh Lu and Ahmed Zeriahi

    Topics Analytic minimal model program, Kähler–Ricci flow, Weak & singular flows · Same authors V. Guedj, A. Zeriahi · Shared terms “parabolic complex”, “degenerate parabolic”, “complex Monge”

  2. The pluripotential Cauchy-Dirichlet problem for complex Monge-Ampere flows

    Vincent Guedj, Hoang Chinh Lu and Ahmed Zeriahi

    Topics Analytic minimal model program, Kähler geometry · Same authors V. Guedj, A. Zeriahi · Shared terms “complex Monge”, “Monge Ampère”, “parabolic complex”

  3. Convergence of scalar curvature of Kahler-Ricci flow on manifolds of positive Kodaira dimension

    Wangjian Jian

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “positive Kodaira”, “Kodaira dimension”, “normalized Kähler”

  4. Kähler-Ricci flows coming out of metric spaces

    Alix Deruelle, Vincent Guedj, Henri Guenancia et al.

    Topics Kähler–Ricci flow, Weak & singular flows, Kähler geometry · Same authors V. Guedj, A. Zeriahi

  5. Stability of solutions to complex Monge-Ampere flows

    Vincent Guedj, Hoang Chinh Lu and Ahmed Zeriahi

    Topics Kähler geometry · Same authors V. Guedj, A. Zeriahi · Shared terms “complex Monge”, “Monge Ampère”, “parabolic complex”

  6. Convergence of the weak Kähler-Ricci Flow on manifolds of general type

    Tat Dat Tô

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “weak Kähler”, “normalized Kähler”, “complex Monge”

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

    Lei Zhang and Zhenlei Zhang

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

  8. On the Ricci curvature of Kahler-Ricci Flow

    Cheuk Yan Fung

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

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 whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  3. 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.