Paper · Apr 25, 2019

A "boundedness implies convergence" principle and its applications to collapsing estimates in Kähler geometry

Wangjian Jian, Yalong Shi

math.DGarXiv:1904.11261
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Abstract

We establish a general "boundedness implies convergence" principle for a family of evolving Riemannian metrics. We then apply this principle to collapsing Calabi-Yau metrics and normalized Kähler-Ricci flows on torus fibered minimal models to obtain convergence results.

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Submitted
Apr 25, 2019
Subjects
math.DG

Comments: All comments are welcome!

Ricci flowKähler–Ricci flowRicci-flat & Calabi–Yau(Non)collapsingKähler geometryAnalytic 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 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 Convergence of scalar curvature of Kahler-Ricci flow on manifolds of positive Kodaira dimension (Wangjian Jian, 2018)1Jian 2018 Diameter and Ricci curvature estimates for long-time solutions of the Kahler-Ricci flow (Wangjian Jian and Jian Song, 2021)8Jian 2021 On the Hölder estimate of Kähler-Ricci flow (Jianchun Chu and Man-Chun Lee, 2021)5Chu 2021 L^∞ estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation (Wangjian Jian and Yalong Shi, 2023)2Jian 2023 Collapsing immortal Kähler-Ricci flows (Hans-Joachim Hein, Man-Chun Lee and Valentino Tosatti, 2024)6Hein 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)7Jian 2026 Type I estimates for the Kahler-Ricci flow I (Wangjian Jian and Jian Song, 2026)4Jian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Convergence of scalar curvature of Kahler-Ricci flow on manifolds of positive Kodaira dimension

    Wangjian Jian

    Topics Analytic minimal model program, Ricci-flat & Calabi–Yau, Kähler–Ricci flow · Same author Wangjian Jian · Shared terms “normalized Kähler”

  2. estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation

    Wangjian Jian and Yalong Shi

    Topics Kähler–Ricci flow, Kähler geometry · Same authors Wangjian Jian, Yalong Shi · Shared terms “normalized Kähler”

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

    Wangjian Jian and Jian Song

    Topics Analytic minimal model program, Kähler–Ricci flow, (Non)collapsing · Same author Wangjian Jian

  4. Type I estimates for the Kahler-Ricci flow I

    Wangjian Jian and Jian Song

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Same author Wangjian Jian

  5. 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 · Shared terms “Yau metrics”

  6. 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 · Shared terms “Yau metrics”

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

    Wangjian Jian and Jian Song

    Topics Analytic minimal model program, Kähler–Ricci flow, (Non)collapsing · Same author Wangjian Jian

  8. 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, Kähler geometry · Same author Wangjian Jian · Shared terms “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) 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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

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.