Paper · Jun 23, 2017

Infinite-time singularity type of the Kähler-Ricci flow

Yashan Zhang

math.DGarXiv:1706.07743v2
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Abstract

For the Kähler-Ricci flow on a compact Kähler manifold with semi-ample canonical line bundle, we prove the singularity type at infinity does not depend on the choice of the initial metric. We also provide new simple proofs for some existing classification results on infinite-time singularity type of the Kähler-Ricci flow.

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Submitted
Jun 23, 2017
Last revised
Oct 16, 2017 (v2)
Subjects
math.DG

Comments: V2, modified Abstract, refined Introduction section, updated references; accepted by The Journal of Geometric Analysis

Ricci flowKähler–Ricci flowSingularitiesKähler geometryAnalytic 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 A note on conical Kähler-Ricci flow on minimal elliptic Kähler surfaces (Yashan Zhang, 2016)5Zhang 2016 Infinite-time singularity type of the Kähler-Ricci flow II (Yashan Zhang, 2018)1Zhang 2018 Local curvature estimates of long-time solutions to the Kähler-Ricci flow (Frederick Tsz-Ho Fong and Yashan Zhang, 2019)2Fong 2019 Integral inequalities for holomorphic maps and applications (Yashan Zhang, 2019)7Zhang 2019 Curvature Estimates for the Continuity Method (Hosea Wondo, 2021)6Wondo 2021 On the Ricci curvature of Kahler-Ricci Flow (Cheuk Yan Fung, 2021)4Fung 2021 Independence of Singularity Type for Numerically Effective Kähler-Ricci Flows (Hosea Wondo and Zhou Zhang, 2023)3Wondo 2023 Type I estimates for the Kahler-Ricci flow I (Wangjian Jian and Jian Song, 2026)8Jian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Infinite-time singularity type of the Kähler-Ricci flow II

    Yashan Zhang

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Same author Yashan Zhang · Shared terms “singularity type”, “infinite time”, “time singularity”

  2. 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, Kähler geometry · Same author Yashan Zhang · Shared terms “semi ample”, “ample canonical”, “canonical line”

  3. Independence of Singularity Type for Numerically Effective Kähler-Ricci Flows

    Hosea Wondo and Zhou Zhang

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “singularity type”, “initial metric”, “semi ample”

  4. 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”

  5. A note on conical Kähler-Ricci flow on minimal elliptic Kähler surfaces

    Yashan Zhang

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Same author Yashan Zhang · Shared terms “canonical line”, “line bundle”

  6. 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”

  7. Integral inequalities for holomorphic maps and applications

    Yashan Zhang

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Yashan Zhang · Shared terms “singularity type”, “infinite time”, “time singularity”

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

    Wangjian Jian and Jian Song

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “time singularity”, “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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.