Paper · Oct 31, 2016

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

Yashan Zhang

math.DGarXiv:1610.09880v2
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Abstract

We prove that, under a semi-ampleness type assumption on the twisted canonical line bundle, the conical Kähler-Ricci flow on a minimal elliptic Kähler surface converges in the sense of currents to a generalized conical Kähler-Einstein on its canonical model. Moreover, the convergence takes place smoothly outside the singular fibers and the chosen divisor.

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Submitted
Oct 31, 2016
Last revised
Aug 11, 2017 (v2)
Subjects
math.DG

Comments: V2; accepted by Acta Mathematica Scientia, English Series

Ricci flowKähler–Ricci flowEinstein metricsKähler geometryKähler–Einstein & K-stabilityAnalytic 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 Collapsing limits of the Kähler-Ricci flow and the continuity method (Yashan Zhang, 2017)7Zhang 2017 Cusp Kähler-Ricci flow on compact Kähler manifold (Jiawei Liu and Xi Zhang, 2017)6Liu 2017 Infinite-time singularity type of the Kähler-Ricci flow (Yashan Zhang, 2017)4Zhang 2017 On a twisted conical Kähler-Ricci flow (Yashan Zhang, 2017)2Zhang 2017 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)3Fong 2019 Stability of the conical Kähler-Ricci flows on Fano manifolds (Jiawei Liu and Xi Zhang, 2019)5Liu 2019 On the Ricci curvature of Kahler-Ricci Flow (Cheuk Yan Fung, 2021)8Fung 2021
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 “singular fibers”, “canonical line”, “line bundle”

  2. On a twisted conical Kähler-Ricci flow

    Yashan Zhang

    Topics Kähler–Ricci flow, Kähler geometry · Same author Yashan Zhang · Shared terms “conical Kähler”

  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, Kähler geometry · Same author Yashan Zhang · Shared terms “singular fibers”, “canonical line”, “line bundle”

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

    Yashan Zhang

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

  5. Stability of the conical Kähler-Ricci flows on Fano manifolds

    Jiawei Liu and Xi Zhang

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “conical Kähler”

  6. Cusp Kähler-Ricci flow on compact Kähler manifold

    Jiawei Liu and Xi Zhang

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “conical Kähler”, “twisted canonical”

  7. Collapsing limits of the Kähler-Ricci flow and the continuity method

    Yashan Zhang

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Same author Yashan Zhang

  8. On the Ricci curvature of Kahler-Ricci Flow

    Cheuk Yan Fung

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

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) 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. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  3. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.