Paper · Oct 16, 2026

Type I estimates for the Kähler-Ricci flow II

Wangjian Jian, Jian Song

math.DGarXiv:2610.22941v2
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Abstract

In this paper, we establish geometric consequences of our Type I scalar curvature estimate for finite-time solutions of the Kähler-Ricci flow on compact Kähler manifolds. We prove a uniform global diameter bound and a Type I diameter bound for every fibre of finite time holomorphic contraction. These estimates yield Gromov-Hausdorff convergence to a unique compact metric space homeomorphic to the normal Kähler space determined by the limiting cohomology class. We also establish a local partial estimate for the normalized flow and show that the tangent shrinkers are normal complex analytic varieties with log terminal singularities.

Comments: comments are welcome

Ricci flowKähler–Ricci flowRicci solitonsSingularitiesKähler geometryScalar curvatureLimit spaces & RCDAnalytic 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 Diameter estimates in Kähler geometry (Bin Guo, Duong H. Phong, Jian Song et al., 2022)7Guo 2022 Finite time singularities of the Kähler-Ricci flow (Wangjian Jian, Jian Song and Gang Tian, 2023)2Jian 2023 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)4Hallgren 2023 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)6Jian 2023 Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces (Ye-Won Luke Cho and Young-Jun Choi, 2024)8Cho 2024 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles (Wangjian Jian and Jian Song, 2026)3Jian 2026 Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II (Wangjian Jian and Jian Song, 2026)5Jian 2026 Type I estimates for the Kahler-Ricci flow I (Wangjian Jian and Jian Song, 2026)1Jian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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 · Same authors Wangjian Jian, Jian Song · Shared terms “finite time”, “compact Kähler”

  2. Finite time singularities of the Kähler-Ricci flow

    Wangjian Jian, Jian Song and Gang Tian PDF

    Topics Kähler–Ricci flow, Limit spaces & RCD, Scalar curvature · Same authors Wangjian Jian, Jian Song · Shared terms “diameter bound”, “time solution”, “finite time”

  3. 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 “finite time”, “diameter”, “limiting”

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

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

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same authors Wangjian Jian, Jian Song · Shared terms “Gromov Hausdorff”, “limiting”

  5. 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 “finite time”, “diameter”, “limiting”

  6. 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 “diameter”, “time solution”, “finite time”

  7. Diameter estimates in Kähler geometry

    Bin Guo, Duong H. Phong, Jian Song et al. PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Jian Song · Shared terms “diameter bound”, “time solution”, “finite time”

  8. Continuity of solutions to complex Monge-Ampère equations on compact Kähler spaces

    Ye-Won Luke Cho and Young-Jun Choi PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “Kähler space”, “normal Kähler”, “terminal singularities”

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) 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)

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.