Paper · May 4, 2021

On the Hölder estimate of Kähler-Ricci flow

Jianchun Chu, Man-Chun Lee

math.DGarXiv:2105.01602
View PDF View HTML arXiv page

As on arXiv

Abstract

In this work, we study the H\"older regularity of the Kähler- Ricci flow on compact Kähler manifolds with semi-ample canonical line bundle. By adapting the method in the work of Hein-Tosatti on collapsing Calabi-Yau metrics, we obtain a uniform spatial Hölder estimate of the Kähler-Ricci flow for all time.

View PDF View HTML arXiv page
Submitted
May 4, 2021
Subjects
math.DG

Comments: 15 pages

Ricci flowKähler–Ricci flowRicci-flat & Calabi–Yau(Non)collapsingKähler geometryAnalytic minimal model program

Its neighbourhood

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 Infinite-time singularity type of the Kähler-Ricci flow (Yashan Zhang, 2017)8Zhang 2017 Convergence of scalar curvature of Kahler-Ricci flow on manifolds of positive Kodaira dimension (Wangjian Jian, 2018)4Jian 2018 Infinite-time singularity type of the Kähler-Ricci flow II (Yashan Zhang, 2018)6Zhang 2018 A "boundedness implies convergence" principle and its applications to collapsing estimates in Kähler geometry (Wangjian Jian and Yalong Shi, 2019)3Jian 2019 Higher-Order Estimates of Long-Time Solutions to the Kähler-Ricci Flow (Frederick Tsz-Ho Fong and Man-Chun Lee, 2020)1Fong 2020 Curvature Estimates for the Continuity Method (Hosea Wondo, 2021)7Wondo 2021 On the Ricci curvature of Kahler-Ricci Flow (Cheuk Yan Fung, 2021)5Fung 2021 Collapsing immortal Kähler-Ricci flows (Hans-Joachim Hein, Man-Chun Lee and Valentino Tosatti, 2024)2Hein 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. 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, 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 “Yau metrics”

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

    Wangjian Jian and Yalong Shi

    Topics Analytic minimal model program, Ricci-flat & Calabi–Yau, Kähler–Ricci flow · Shared terms “Yau metrics”

  4. 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 · Shared terms “ample canonical”, “semi ample”, “canonical line”

  5. On the Ricci curvature of Kahler-Ricci Flow

    Cheuk Yan Fung

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

  6. Infinite-time singularity type of the Kähler-Ricci flow II

    Yashan Zhang

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

  7. Curvature Estimates for the Continuity Method

    Hosea Wondo

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

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

    Yashan Zhang

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

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.