Paper · Jun 24, 2021

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

Kyle Broder

math.DGarXiv:2106.13343v3
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Abstract

We show that the singularities of the twisted Kähler–Einstein metric arising as the long-time solution of the Kähler–Ricci flow or in the collapsed limit of Ricci-flat Kähler metrics is intimately related to the holomorphic sectional curvature of the reference conical geometry. This provides an alternative proof of the second-order estimate obtained by Gross–Tosatti–Zhang with explicit constants appearing in the divisorial pole.

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Submitted
Jun 24, 2021
Last revised
Sep 13, 2022 (v3)
Subjects
math.DG, math.CV

Comments: 16 pages, correction of mistakes in the previous version, comments are welcome

Ricci flowKähler–Ricci flowEinstein metricsRicci-flat & Calabi–YauSingularitiesKähler geometryLong-time existence & convergenceKähler–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 The Kähler-Ricci flow on manifolds with negative holomorphic curvature (Freid Tong, 2018)8Tong 2018 Longtime existence of Kähler Ricci flow and holomorphic sectional curvature (Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al., 2018)6Huang 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)2Song 2019 Complete Kähler-Einstein metric on Stein manifolds with negative curvature (Man-Chun Lee, 2019)7Lee 2019 Geometry of twisted Kähler-Einstein metrics and collapsing (Mark Gross, Valentino Tosatti and Yuguang Zhang, 2019)4Gross 2019 Higher-Order Estimates of Long-Time Solutions to the Kähler-Ricci Flow (Frederick Tsz-Ho Fong and Man-Chun Lee, 2020)3Fong 2020 Twisted Kähler-Einstein Metrics and Collapsing (Kyle Broder, 2020)1Broder 2020 Remarks on Singular Kähler-Einstein Metrics (Max Hallgren and Gábor Székelyhidi, 2025)5Hallgren 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Twisted Kähler-Einstein Metrics and Collapsing

    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”

  2. 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”

  3. Higher-Order Estimates of Long-Time Solutions to the Kähler-Ricci Flow

    Frederick Tsz-Ho Fong and Man-Chun Lee

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

  4. 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”, “collapsed”, “holomorphic”

  5. Remarks on Singular Kähler-Einstein Metrics

    Max Hallgren and Gábor Székelyhidi

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

  6. Longtime existence of Kähler Ricci flow and holomorphic sectional curvature

    Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al.

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

  7. Complete Kähler-Einstein metric on Stein manifolds with negative curvature

    Man-Chun Lee

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

  8. The Kähler-Ricci flow on manifolds with negative holomorphic curvature

    Freid Tong

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

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. 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.