Paper · Mar 31, 2020

Twisted Kähler-Einstein Metrics and Collapsing

Kyle Broder

math.DGarXiv:2003.14009v3
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Abstract

We extend the arguments of Tosatti-Zhang to reduce a well-known conjecture concerning the structure of the Gromov-Hausdorff limit in both the setting of degenerating Calabi-Yau manifolds and the Kähler-Ricci flow to a certain partial second-order estimate.

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Submitted
Mar 31, 2020
Last revised
Jun 29, 2021 (v3)
Subjects
math.DG, math.CV

Comments: Updated based on reviewers comments

Ricci flowKähler–Ricci flowEinstein metricsRicci-flat & Calabi–Yau(Non)collapsingKähler geometryLimit spaces & RCDKähler–Einstein & K-stability

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 behavior of Ricci-flat Kahler metrics and long time solutions of the Kahler-Ricci flow (Jian Song, Gang Tian and Zhenlei Zhang, 2019)4Song 2019 Geometry of twisted Kähler-Einstein metrics and collapsing (Mark Gross, Valentino Tosatti and Yuguang Zhang, 2019)1Gross 2019 Second-order estimates for collapsed limits of Ricci-flat Kähler metrics (Kyle Broder, 2021)2Broder 2021 On the collapsing of Calabi-Yau manifolds and Kähler-Ricci flows (Yang Li and Valentino Tosatti, 2021)3Li 2021 Remarks on Singular Kähler-Einstein Metrics (Max Hallgren and Gábor Székelyhidi, 2025)5Hallgren 2025 Fano Fibrations and Twisted Kähler-Einstein Metrics I (Alexander Bednarek, 2025)7Bednarek 2025 Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow (Alexander Bednarek, 2025)8Bednarek 2025 Gromov-Hausdorff limits of immortal Kähler-Ricci flows (Man-Chun Lee, Valentino Tosatti and Junsheng Zhang, 2026)6Lee 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Geometry of twisted Kähler-Einstein metrics and collapsing

    Mark Gross, Valentino Tosatti and Yuguang Zhang

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

  2. Second-order estimates for collapsed limits of Ricci-flat Kähler metrics

    Kyle Broder

    Topics Ricci-flat & Calabi–Yau, Kähler–Einstein & K-stability, Kähler–Ricci flow · Same author Kyle Broder · Shared terms “order estimate”, “twisted Kähler”, “Tosatti”

  3. 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 “Hausdorff limit”, “Gromov Hausdorff”

  4. Collapsing behavior of Ricci-flat Kahler metrics and long time solutions of the Kahler-Ricci flow

    Jian Song, Gang Tian and Zhenlei Zhang

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

  5. Remarks on Singular Kähler-Einstein Metrics

    Max Hallgren and Gábor Székelyhidi

    Topics Ricci-flat & Calabi–Yau, Kähler–Einstein & K-stability, Kähler–Ricci flow

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

    Man-Chun Lee, Valentino Tosatti and Junsheng Zhang

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “twisted Kähler”, “Hausdorff limit”, “Gromov Hausdorff”

  7. Fano Fibrations and Twisted Kähler-Einstein Metrics I

    Alexander Bednarek

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

  8. Fano Fibrations and Twisted Kähler-Einstein Metrics II: The Kähler-Ricci Flow

    Alexander Bednarek

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “twisted 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) 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.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.