Paper · Sep 30, 2026

Type I estimates for the Kahler-Ricci flow I

Wangjian Jian, Jian Song

math.DGarXiv:2609.39620
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Abstract

We establish the Type I estimate for the scalar curvature of the Kahler-Ricci flow on compact Kahler manifolds, for arbitrary smooth initial metrics, whenever the flow develops finite time singularities. This estimate is a building block for the analytic minimal model program with Ricci flow.

Ricci flowKähler–Ricci flowSingularitiesKähler 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 A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds (Wangjian Jian, Jian Song and Gang Tian, 2023)4Jian 2023 Finite time singularities of the Kähler-Ricci flow (Wangjian Jian, Jian Song and Gang Tian, 2023)5Jian 2023 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)6Hallgren 2023 Expanding Soliton Models for Kähler-Ricci Flow Near Conical Singularities (Longteng Chen, Max Hallgren and Lucas Lavoyer, 2026)8Chen 2026 Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program (Longteng Chen, Max Hallgren and Lucas Lavoyer, 2026)7Chen 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles (Wangjian Jian and Jian Song, 2026)1Jian 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II (Wangjian Jian and Jian Song, 2026)2Jian 2026 Type I estimates for the Kähler-Ricci flow II (Wangjian Jian and Jian Song, 2026)3Jian 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 Singularities of the Kähler–Ricci Flow on Fano Bundles

    Wangjian Jian and Jian Song PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Same authors Wangjian Jian, Jian Song · Shared terms “time singularity”, “finite time”

  2. Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II

    Wangjian Jian and Jian Song PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Scalar curvature · Same authors Wangjian Jian, Jian Song · Shared terms “time singularity”, “finite time”

  3. Type I estimates for the Kähler-Ricci flow II

    Wangjian Jian and Jian Song PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Scalar curvature · Same authors Wangjian Jian, Jian Song · Shared terms “finite time”, “compact Kähler”

  4. A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds

    Wangjian Jian, Jian Song and Gang Tian PDF

    Topics Kähler–Ricci flow, Scalar curvature, Kähler geometry · Same authors Wangjian Jian, Jian Song · Shared terms “finite time”

  5. Finite time singularities of the Kähler-Ricci flow

    Wangjian Jian, Jian Song and Gang Tian PDF

    Topics Kähler–Ricci flow, Scalar curvature, Kähler geometry · Same authors Wangjian Jian, Jian Song · Shared terms “time singularity”, “finite time”

  6. Geometric regularity of blow-up limits of the Kähler-Ricci flow

    Max Hallgren, Wangjian Jian, Jian Song et al. PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same authors Wangjian Jian, Jian Song

  7. Kähler-Ricci Tangent Flows in the Analytic Minimal Model Program

    Longteng Chen, Max Hallgren and Lucas Lavoyer PDF

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

  8. Expanding Soliton Models for Kähler-Ricci Flow Near Conical Singularities

    Longteng Chen, Max Hallgren and Lucas Lavoyer PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, 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) 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.