Paper · Jan 30, 2020

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

Frederick Tsz-Ho Fong, Man-Chun Lee

math.DGarXiv:2001.11555
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Abstract

In this article, we study the higher-order regularity of the Kähler-Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle. We proved, using a parabolic analogue of Hein-Tosatti's work on collapsing Calabi-Yau metrics, that when the generic fibers of the Iitaka fibration are biholomorphic to each other, the flow converges locally smoothly away from singular fibers to a negative Kähler-Einstein metric on the base manifold. In particular, we proved that the Ricci curvature of the flow is uniformly bounded on any compact subsets away from singular fibers when the generic fibers are biholomorphic to each other.

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Submitted
Jan 30, 2020
Subjects
math.DG

Comments: 30 pages

Ricci flowKähler–Ricci flowEinstein metricsRicci-flat & Calabi–Yau(Non)collapsingKähler geometryLong-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 Local curvature estimates of long-time solutions to the Kähler-Ricci flow (Frederick Tsz-Ho Fong and Yashan Zhang, 2019)3Fong 2019 Collapsing behavior of Ricci-flat Kahler metrics and long time solutions of the Kahler-Ricci flow (Jian Song, Gang Tian and Zhenlei Zhang, 2019)5Song 2019 Complete Kähler-Einstein metric on Stein manifolds with negative curvature (Man-Chun Lee, 2019)7Lee 2019 On the Hölder estimate of Kähler-Ricci flow (Jianchun Chu and Man-Chun Lee, 2021)1Chu 2021 On the Ricci curvature of Kahler-Ricci Flow (Cheuk Yan Fung, 2021)4Fung 2021 Collapsing immortal Kähler-Ricci flows (Hans-Joachim Hein, Man-Chun Lee and Valentino Tosatti, 2024)2Hein 2024 Gromov-Hausdorff limits of immortal Kähler-Ricci flows (Man-Chun Lee, Valentino Tosatti and Junsheng Zhang, 2026)8Lee 2026 Gromov-Hausdorff Convergence of Constant Scalar Curvature Kähler Metrics near the Canonical Class (Frederick Tsz-Ho Fong and Junbang Liu, 2026)6Fong 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Hölder estimate of Kähler-Ricci flow

    Jianchun Chu and Man-Chun Lee

    Topics Analytic minimal model program, Ricci-flat & Calabi–Yau, Kähler–Ricci flow · Same author Man-Chun Lee · Shared terms “Hein Tosatti”, “ample canonical”, “Yau metrics”

  2. Collapsing immortal Kähler-Ricci flows

    Hans-Joachim Hein, Man-Chun Lee and Valentino Tosatti

    Topics Analytic minimal model program, Ricci-flat & Calabi–Yau, Kähler–Ricci flow · Same author Man-Chun Lee · Shared terms “singular fibers”, “Iitaka fibration”, “Yau metrics”

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

    Frederick Tsz-Ho Fong and Yashan Zhang

    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. On the Ricci curvature of Kahler-Ricci Flow

    Cheuk Yan Fung

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

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

    Jian Song, Gang Tian and Zhenlei Zhang

    Topics Analytic minimal model program, Ricci-flat & Calabi–Yau, Kähler–Einstein & K-stability · Shared terms “semi ample”, “ample canonical”, “uniformly bounded”

  6. 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–Einstein & K-stability, Kähler–Ricci flow · Same author Frederick Tsz-Ho Fong · Shared terms “ample canonical”, “semi ample”, “Tosatti”

  7. Complete Kähler-Einstein metric on Stein manifolds with negative curvature

    Man-Chun Lee

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Same author Man-Chun Lee · Shared terms “negative Kähler”

  8. 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 · Same author Man-Chun Lee · 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) 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.