Paper · May 21, 2018

Convergence of scalar curvature of Kahler-Ricci flow on manifolds of positive Kodaira dimension

Wangjian Jian

math.DGarXiv:1805.07884
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Abstract

In this paper, we consider Kahler-Ricci flow on n-dimensional Kahler manifold with semi-ample canonical line bundle and 0< m:= Kod(X)<n. Such manifolds admit a Calabi-Yau fibration over its canonical model. We prove that the scalar curvature of the Kahler metric along the normalized Kahler-Ricci flow converge to -m outside the singular set of this fibration.

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Submitted
May 21, 2018
Subjects
math.DG

Comments: 23 pages. All comments are welcome

Ricci flowKähler–Ricci flowRicci-flat & Calabi–YauKähler geometryScalar curvatureAnalytic 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 Convergence of weak Kähler-Ricci Flows on minimal models of positive Kodaira dimension (P. Eyssidieux, V. Guedj and A. Zeriahi, 2016)3Eyssidieux 2016 Infinite-time singularity type of the Kähler-Ricci flow II (Yashan Zhang, 2018)5Zhang 2018 Collapsing behavior of Ricci-flat Kahler metrics and long time solutions of the Kahler-Ricci flow (Jian Song, Gang Tian and Zhenlei Zhang, 2019)8Song 2019 A "boundedness implies convergence" principle and its applications to collapsing estimates in Kähler geometry (Wangjian Jian and Yalong Shi, 2019)7Jian 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)2Fung 2021 Convergence of Scalar Curvature of Long Time Kähler-Ricci Flow on Kähler Manifold (Lei Zhang and Zhenlei Zhang, 2026)4Zhang 2026 Type I estimates for the Kahler-Ricci flow I (Wangjian Jian and Jian Song, 2026)6Jian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  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, Scalar curvature · Same author Wangjian Jian · Shared terms “Kodaira dimension”, “normalized Kähler”, “canonical model”

  2. On the Ricci curvature of Kahler-Ricci Flow

    Cheuk Yan Fung

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

  3. Convergence of weak Kähler-Ricci Flows on minimal models of positive Kodaira dimension

    P. Eyssidieux, V. Guedj and A. Zeriahi

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “positive Kodaira”, “Kodaira dimension”, “normalized Kähler”

  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, Scalar curvature · Shared terms “Kodaira dimension”, “normalized Kähler”, “flow converges”

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

    Yashan Zhang

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

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

    Wangjian Jian and Jian Song

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

  7. 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 · Same author Wangjian Jian · Shared terms “normalized Kähler”

  8. 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–Ricci flow · Shared terms “Kodaira dimension”, “canonical model”, “semi ample”

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) 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. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  3. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.