Paper · May 3, 2019

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

Tat Dat Tô

math.CVarXiv:1905.01276v3
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Abstract

We study the Kähler-Ricci flow on compact Kähler manifolds whose canonical bundle is big. We show that the normalized Kähler-Ricci flow has long time existence in the viscosity sense, is continuous in a Zariski open set, and converges to the unique singular Kähler-Einstein metric in the canonical class. The key ingredient is a viscosity theory for degenerate complex Monge-Ampère flows in big classes that we develop, extending and refining the approach of Eyssidieux-Guedj-Zeriahi.

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Submitted
May 3, 2019
Last revised
Nov 10, 2023 (v3)
Subjects
math.CV, math.AP, math.DG
Journal
Int. Math. Res. Not. 2021 No. 8, 6373-6404
DOI
10.1093/imrn/rnz256

Comments: details added in proof of Theorem 4.3

Ricci flowKähler–Ricci flowEinstein metricsKähler geometryLong-time existence & convergenceKähler–Einstein & K-stabilityAnalytic minimal model program

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 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 weak Kähler-Ricci Flows on minimal models of positive Kodaira dimension (P. Eyssidieux, V. Guedj and A. Zeriahi, 2016)5Eyssidieux 2016 Cusp Kähler-Ricci flow on compact Kähler manifold (Jiawei Liu and Xi Zhang, 2017)4Liu 2017 Pluripotential Monge-Ampère flows in Big Cohomology Classes (Quang-Tuan Dang, 2021)3Dang 2021 On the Ricci curvature of Kahler-Ricci Flow (Cheuk Yan Fung, 2021)8Fung 2021 Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces (Ye-Won Luke Cho and Young-Jun Choi, 2024)1Cho 2024 On Kähler-Einstein Currents (Yifan Chen, Shih-Kai Chiu, Max Hallgren et al., 2025)2Chen 2025 Gromov-Hausdorff limits of immortal Kähler-Ricci flows (Man-Chun Lee, Valentino Tosatti and Junsheng Zhang, 2026)6Lee 2026 Scalar Curvature, Volumes and the Bergman Kernel (Bo-Yong Chen, Yuanpu Xiong and Liyou Zhang, 2026)7Chen 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces

    Ye-Won Luke Cho and Young-Jun Choi

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “weak Kähler”, “complex Monge”, “Monge Ampère”

  2. On Kähler-Einstein Currents

    Yifan Chen, Shih-Kai Chiu, Max Hallgren et al.

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Einstein metrics · Same author Tat Dat Tô · Shared terms “singular Kähler”

  3. Pluripotential Monge-Ampère flows in Big Cohomology Classes

    Quang-Tuan Dang

    Topics Analytic minimal model program, Kähler geometry · Shared terms “Ampere flow”, “Monge Ampère”, “compact Kähler”

  4. Cusp Kähler-Ricci flow on compact Kähler manifold

    Jiawei Liu and Xi Zhang

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “compact Kähler”, “canonical bundle”

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

    P. Eyssidieux, V. Guedj and A. Zeriahi

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

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

    Man-Chun Lee, Valentino Tosatti and Junsheng Zhang

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “normalized Kähler”, “canonical bundle”, “compact Kähler”

  7. Scalar Curvature, Volumes and the Bergman Kernel

    Bo-Yong Chen, Yuanpu Xiong and Liyou Zhang

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “singular Kähler”, “compact Kähler”

  8. On the Ricci curvature of Kahler-Ricci Flow

    Cheuk Yan Fung

    Topics Analytic minimal model program, Kähler–Ricci flow, Einstein metrics

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) 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) 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)

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.