Paper · Jun 16, 2026

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

Lei Zhang, Zhenlei Zhang

math.DGarXiv:2606.17402
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Abstract

This paper is concerned with a class of the long time Kähler-Ricci flow on a compact Kähler manifold. It is shown that the uniform -entropy or uniform Sobolev inequality along the normalized Kähler-Ricci flow with semiample canonical bundle. As a consequence, we prove that the scalar curvature of the Kähler metrics along the normalized Kähler-Ricci flow converge to negative Kodaira dimension of the compact Kähler manifold.

Comments: 20 pages, all comments are welcome

Ricci flowKähler–Ricci flowKähler geometryScalar curvatureAnalytic 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 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 the Ricci curvature of Kahler-Ricci Flow (Cheuk Yan Fung, 2021)4Fung 2021 Collapsing immortal Kähler-Ricci flows (Hans-Joachim Hein, Man-Chun Lee and Valentino Tosatti, 2024)7Hein 2024 Gromov-Hausdorff limits of immortal Kähler-Ricci flows (Man-Chun Lee, Valentino Tosatti and Junsheng Zhang, 2026)5Lee 2026 Higher-order Ricci estimates along immortal Kähler-Ricci flows (Wenrui Kong, 2026)6Kong 2026 Finite Time Type I Singularities of the Kähler Ricci Flow (Tongxin Xu and Zhenlei Zhang, 2026)1Xu 2026 Type I estimates for the Kahler-Ricci flow I (Wangjian Jian and Jian Song, 2026)3Jian 2026 Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits (Lei Zhang and Zhenlei Zhang, 2026)2Zhang 2026 Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class (Frederick Tsz-Ho Fong and Junbang Liu, 2026)8Fong 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 Type I Singularities of the Kähler Ricci Flow

    Tongxin Xu and Zhenlei Zhang PDF

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

  2. Finite Time Noncollapsing Kähler-Ricci Flow: Gaussian Type Heat Kernel, Gromov-Hausdorff Limits

    Lei Zhang and Zhenlei Zhang PDF

    Topics Kähler–Ricci flow, Scalar curvature, Kähler geometry · Same authors Lei Zhang, Zhenlei Zhang · Shared terms “compact Kähler”

  3. Type I estimates for the Kahler-Ricci flow I

    Wangjian Jian and Jian Song PDF

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

  4. On the Ricci curvature of Kahler-Ricci Flow

    Cheuk Yan Fung PDF

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

  5. 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, Kähler geometry · Shared terms “normalized Kähler”, “semiample canonical”, “canonical bundle”

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

    Wenrui Kong PDF

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

  7. Collapsing immortal Kähler-Ricci flows

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

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

  8. Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class

    Frederick Tsz-Ho Fong and Junbang Liu PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Scalar curvature · Shared terms “normalized Kähler”, “Kähler metrics”, “canonical bundle”

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979)

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. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  3. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.