Paper · May 3, 2017

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

Yashan Zhang

math.DGarXiv:1705.01434v3
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Abstract

We consider the Kähler-Ricci flow on certain Calabi-Yau fibration, which is a Calabi-Yau fibration with one dimensional base or a product of two Calabi-Yau fibrations with one dimensional bases. Assume the Kähler-Ricci flow on total space admits a uniform lower bound for Ricci curvature, then the flow converges in Gromov-Hausdorff topology to the metric completion of the regular part of generalized Kähler-Einstein current on the base, which is a compact length metric space homeomorphic to the base. The analogue results for the continuity method on such Calabi-Yau fibrations are also obtained. Moreover, we show the continuity method starting from a suitable Kähler metric on the total space of a Fano fibration with one dimensional base converges in Gromov-Hausdorff topology to a compact metric on the base. During the proof, we show the metric completion of the regular part of a generalized Kähler-Einstein current on a Riemann surface is compact.

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Submitted
May 3, 2017
Last revised
Apr 22, 2018 (v3)
Subjects
math.DG

Comments: V3, some incorrect sentences removed, typos fixed, references updated; accepted by Mathematische Annalen

Ricci flowKähler–Ricci flowEinstein metricsRicci-flat & Calabi–Yau(Non)collapsingSurfacesKähler geometryLimit spaces & RCDKähler–Einstein & K-stabilityAnalytic minimal model program

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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.

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  1. 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–Einstein & K-stability · Shared terms “metric completion”, “regular part”, “Gromov Hausdorff”

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

    Yashan Zhang

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

  3. Geometry of twisted Kähler-Einstein metrics and collapsing

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    Topics Analytic minimal model program, Ricci-flat & Calabi–Yau, Kähler–Einstein & K-stability · Shared terms “Gromov Hausdorff”

  4. Gromov-Hausdorff limits of immortal Kähler-Ricci flows

    Man-Chun Lee, Valentino Tosatti and Junsheng Zhang

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “metric completion”, “Gromov Hausdorff”, “Hausdorff topology”

  5. Twisted Kähler-Einstein Metrics and Collapsing

    Kyle Broder

    Topics Ricci-flat & Calabi–Yau, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “Gromov Hausdorff”

  6. On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows

    Yang Li and Valentino Tosatti

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, (Non)collapsing · Shared terms “Gromov Hausdorff”

  7. 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 “fibration”, “Riemann”

  8. Fano Fibrations and Twisted Kähler-Einstein Metrics I

    Alexander Bednarek

    Topics Analytic minimal model program, Kähler–Einstein & K-stability, Kähler–Ricci flow · Shared terms “fibration”, “regular part”, “Fano”

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) 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)

In the guidebook

  1. 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.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.