Paper · Mar 25, 2026

Higher-order Ricci estimates along immortal Kähler-Ricci flows

Wenrui Kong

math.DGarXiv:2603.24380
View PDF View HTML arXiv page

As on arXiv

Abstract

We study higher-order curvature estimates along Kähler-Ricci flows on compact Kähler manifolds of intermediate Kodaira dimension. We prove that away from singular fibers, the Ricci curvature is uniformly bounded in , the Laplacian of the Ricci curvature in , and the scalar curvature in . We identify a geometric obstruction to higher-order curvature bounds, whose non-vanishing causes a specific third-order derivative of the Ricci curvature to blow up at rate . Uniform bounds for every hold for the Ricci curvature in the isotrivial case, and for the full Riemann curvature in the torus-fibered case.

View PDF View HTML arXiv page
Submitted
Mar 25, 2026
Subjects
math.DG, math.AP

Comments: 45 pages

Ricci flowKähler–Ricci flowSingularitiesKähler geometryScalar curvatureLong-time existence & convergenceAnalytic 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 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 Curvature Estimates for the Continuity Method (Hosea Wondo, 2021)7Wondo 2021 On the Ricci curvature of Kahler-Ricci Flow (Cheuk Yan Fung, 2021)3Fung 2021 Collapsing immortal Kähler-Ricci flows (Hans-Joachim Hein, Man-Chun Lee and Valentino Tosatti, 2024)1Hein 2024 Immortal solutions of the Kähler-Ricci flow (Valentino Tosatti, 2024)6Tosatti 2024 Gromov-Hausdorff limits of immortal Kähler-Ricci flows (Man-Chun Lee, Valentino Tosatti and Junsheng Zhang, 2026)4Lee 2026 Convergence of Scalar Curvature of Long Time Kähler-Ricci Flow on Kähler Manifold (Lei Zhang and Zhenlei Zhang, 2026)2Zhang 2026 Finite Time Type I Singularities of the Kähler Ricci Flow (Tongxin Xu and Zhenlei Zhang, 2026)8Xu 2026 Type I estimates for the Kahler-Ricci flow I (Wangjian Jian and Jian Song, 2026)5Jian 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

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

  2. 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, Scalar curvature · Shared terms “Kodaira dimension”, “compact Kähler”

  3. On the Ricci curvature of Kahler-Ricci Flow

    Cheuk Yan Fung

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

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

    Man-Chun Lee, Valentino Tosatti and Junsheng Zhang

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

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

    Wangjian Jian and Jian Song

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

  6. Immortal solutions of the Kähler-Ricci flow

    Valentino Tosatti

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

  7. Curvature Estimates for the Continuity Method

    Hosea Wondo

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

  8. Finite Time Type I Singularities of the Kähler Ricci Flow

    Tongxin Xu and Zhenlei Zhang

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

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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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)

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. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.