Paper · Apr 17, 2019

Collapsing behavior of Ricci-flat Kahler metrics and long time solutions of the Kahler-Ricci flow

Jian Song, Gang Tian, Zhenlei Zhang

math.DGarXiv:1904.08345
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Abstract

We prove a uniform diameter bound for long time solutions of the normalized Kahler-Ricci flow on an -dimensional projective manifold with semi-ample canonical bundle under the assumption that the Ricci curvature is uniformly bounded for all time in a fixed domain containing a fibre of over its canonical model . This assumption on the Ricci curvature always holds when the Kodaira dimension of is , or when the general fibre of over its canonical model is a complex torus. In particular, the normalized Kahler-Ricci flow converges in Gromov-Hausdorff topolopy to its canonical model when has Kodaira dimension with being semi-ample and the general fibre of over its canonical model being a complex torus. We also prove the Gromov-Hausdorff limit of collapsing Ricci-flat Kahler metrics on a holomorphically fibred Calabi-Yau manifold is unique and is homeomorphic to the metric completion of the corresponding twisted Kahler-Einstein metric on the regular part of its base.

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Submitted
Apr 17, 2019
Subjects
math.DG, math.AG, math.AP
Ricci flowKähler–Ricci flowEinstein metricsRicci-flat & Calabi–Yau(Non)collapsingKähler geometryLimit spaces & RCDLong-time existence & convergenceKähler–Einstein & K-stabilityAnalytic minimal model program

Its neighbourhood

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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 2016, 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 Relative volume comparison of Ricci Flow and its applications (Gang Tian and Zhenlei Zhang, 2018)8Tian 2018 Geometry of twisted Kähler-Einstein metrics and collapsing (Mark Gross, Valentino Tosatti and Yuguang Zhang, 2019)3Gross 2019 Diameter and Ricci curvature estimates for long-time solutions of the Kahler-Ricci flow (Wangjian Jian and Jian Song, 2021)1Jian 2021 On the Ricci curvature of Kahler-Ricci Flow (Cheuk Yan Fung, 2021)5Fung 2021 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)6Hallgren 2023 Gromov-Hausdorff limits of immortal Kähler-Ricci flows (Man-Chun Lee, Valentino Tosatti and Junsheng Zhang, 2026)7Lee 2026 Convergence of Scalar Curvature of Long Time Kähler-Ricci Flow on Kähler Manifold (Lei Zhang and Zhenlei Zhang, 2026)4Zhang 2026
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  1. 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, Limit spaces & RCD · Same author Jian Song · Shared terms “canonical model”, “normalized Kähler”, “semi ample”

  2. Finite time singularities of the Kähler-Ricci flow

    Wangjian Jian, Jian Song and Gang Tian

    Topics Kähler–Ricci flow, (Non)collapsing, Limit spaces & RCD · Same authors Jian Song, Gang Tian · Shared terms “time solution”, “diameter bound”, “Gromov Hausdorff”

  3. Geometry of twisted Kähler-Einstein metrics and collapsing

    Mark Gross, Valentino Tosatti and Yuguang Zhang

    Topics Analytic minimal model program, Ricci-flat & Calabi–Yau, Kähler–Einstein & K-stability · Shared terms “twisted Kähler”, “Kodaira dimension”, “canonical bundle”

  4. 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, Kähler geometry · Same author Zhenlei Zhang · Shared terms “normalized Kähler”, “Kodaira dimension”, “long time”

  5. On the Ricci curvature of Kahler-Ricci Flow

    Cheuk Yan Fung

    Topics Analytic minimal model program, Kähler–Ricci flow, Einstein metrics · Shared terms “Kodaira dimension”, “semi ample”, “ample canonical”

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

    Max Hallgren, Wangjian Jian, Jian Song et al.

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

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

    Man-Chun Lee, Valentino Tosatti and Junsheng Zhang

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “canonical model”, “Gromov Hausdorff”, “normalized Kähler”

  8. Relative volume comparison of Ricci Flow and its applications

    Gang Tian and Zhenlei Zhang

    Topics Kähler–Ricci flow, (Non)collapsing, Limit spaces & RCD · Same authors Gang Tian, Zhenlei Zhang · Shared terms “Gromov Hausdorff”

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… 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. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  3. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.