Paper · Mar 14, 2019

Local curvature estimates of long-time solutions to the Kähler-Ricci flow

Frederick Tsz-Ho Fong, Yashan Zhang

math.DGarXiv:1903.05939v2
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Abstract

We study the local curvature estimates of long-time solutions to the normalized Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundles. Using these estimates, we prove that on such a manifold, the set of singular fibers of the semi-ample fibration on which the Riemann curvature blows up at time-infinity is independent of the choice of the initial Kähler metric. Moreover, when the regular fiber of the semi-ample fibration is not a finite quotient of a torus, we determine the exact curvature blow-up rate of the Kähler-Ricci flow near the regular fiber.

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Submitted
Mar 14, 2019
Last revised
Sep 26, 2020 (v2)
Subjects
math.DG

Comments: Small changes; accepted by Advances in Mathematics

Ricci flowKähler–Ricci flowSingularitiesKähler geometryLong-time existence & convergenceAnalytic minimal model program

Its neighbourhood

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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 note on conical Kähler-Ricci flow on minimal elliptic Kähler surfaces (Yashan Zhang, 2016)7Zhang 2016 Infinite-time singularity type of the Kähler-Ricci flow (Yashan Zhang, 2017)2Zhang 2017 Infinite-time singularity type of the Kähler-Ricci flow II (Yashan Zhang, 2018)1Zhang 2018 Higher-Order Estimates of Long-Time Solutions to the Kähler-Ricci Flow (Frederick Tsz-Ho Fong and Man-Chun Lee, 2020)3Fong 2020 Diameter and Ricci curvature estimates for long-time solutions of the Kahler-Ricci flow (Wangjian Jian and Jian Song, 2021)6Jian 2021 Curvature Estimates for the Continuity Method (Hosea Wondo, 2021)4Wondo 2021 On the Ricci curvature of Kahler-Ricci Flow (Cheuk Yan Fung, 2021)5Fung 2021 Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class (Frederick Tsz-Ho Fong and Junbang Liu, 2026)8Fong 2026
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  1. Infinite-time singularity type of the Kähler-Ricci flow II

    Yashan Zhang

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Same author Yashan Zhang · Shared terms “semi ample”, “ample fibration”, “singular fibers”

  2. Infinite-time singularity type of the Kähler-Ricci flow

    Yashan Zhang

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Same author Yashan Zhang · Shared terms “semi ample”, “ample canonical”, “canonical line”

  3. Higher-Order Estimates of Long-Time Solutions to the Kähler-Ricci Flow

    Frederick Tsz-Ho Fong and Man-Chun Lee

    Topics Analytic minimal model program, Kähler–Ricci flow, Long-time existence & convergence · Same author Frederick Tsz-Ho Fong · Shared terms “singular fibers”, “semi ample”, “ample canonical”

  4. Curvature Estimates for the Continuity Method

    Hosea Wondo

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

  5. 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 “semi ample”, “ample canonical”, “canonical line”

  6. Diameter and Ricci curvature estimates for long-time solutions of the Kahler-Ricci flow

    Wangjian Jian and Jian Song

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

  7. A note on conical Kähler-Ricci flow on minimal elliptic Kähler surfaces

    Yashan Zhang

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Same author Yashan Zhang · Shared terms “singular fibers”, “canonical line”, “line bundle”

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

    Frederick Tsz-Ho Fong and Junbang Liu

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Same author Frederick Tsz-Ho Fong · Shared terms “semi ample”, “ample canonical”, “normalized 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 can stop in finite time: curvature blows up, 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 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. 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.