Paper · Sep 20, 2022

Diameter estimates in Kähler geometry

Bin Guo, Duong H. Phong, Jian Song, Jacob Sturm

math.DGarXiv:2209.09428
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Abstract

Diameter estimates for Kähler metrics are established which require only an entropy bound and no lower bound on the Ricci curvature. The proof builds on recent PDE techniques for estimates for the Monge-Ampère equation, with a key improvement allowing degeneracies of the volume form of codimension strictly greater than one. As a consequence, diameter bounds are obtained for long-time solutions of the Kähler-Ricci flow and finite-time solutions when the limiting class is big, as well as for special fibrations of Calabi-Yau manifolds.

Ricci flowKähler–Ricci flowRicci-flat & Calabi–YauSingularitiesKähler geometryLong-time existence & convergence

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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 2021, 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 Flows of G₂-Structures associated to Calabi-Yau Manifolds (Sébastien Picard and Caleb Suan, 2022)5Picard 2022 A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds (Wangjian Jian, Jian Song and Gang Tian, 2023)1Jian 2023 Finite time singularities of the Kähler-Ricci flow (Wangjian Jian, Jian Song and Gang Tian, 2023)6Jian 2023 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)7Hallgren 2023 Complex Monge-Ampère equation for positive (p,p)-forms on compact Kähler manifolds (Mathew George, 2024)4George 2024 Weak transcendental base-point freeness and diameter lower bounds for the Kähler-Ricci flow (Junsheng Zhang, 2025)8Zhang 2025 Parabolic Monge-Ampère equations on a family of Calabi-Yau manifolds (Jiyuan Han, 2026)2Han 2026 Type I estimates for the Kähler-Ricci flow II (Wangjian Jian and Jian Song, 2026)3Jian 2026
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  1. A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds

    Wangjian Jian, Jian Song and Gang Tian PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Jian Song · Shared terms “diameter estimates”, “time solution”, “finite time”

  2. Parabolic Monge-Ampère equations on a family of Calabi-Yau manifolds

    Jiyuan Han PDF

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Kähler geometry · Shared terms “diameter estimates”, “Ampère equation”, “Monge Ampère”

  3. Type I estimates for the Kähler-Ricci flow II

    Wangjian Jian and Jian Song PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Jian Song · Shared terms “diameter bound”, “time solution”, “finite time”

  4. Complex Monge-Ampère equation for positive -forms on compact Kähler manifolds

    Mathew George PDF

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Kähler geometry · Shared terms “Ampère equation”, “Monge Ampère”, “PDE”

  5. Flows of -Structures associated to Calabi-Yau Manifolds

    Sébastien Picard and Caleb Suan PDF

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Long-time existence & convergence · Shared terms “Ampère equation”, “Monge Ampère”

  6. Finite time singularities of the Kähler-Ricci flow

    Wangjian Jian, Jian Song and Gang Tian PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Jian Song · Shared terms “time solution”, “diameter bound”, “finite time”

  7. Geometric regularity of blow-up limits of the Kähler-Ricci flow

    Max Hallgren, Wangjian Jian, Jian Song et al. PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Jian Song · Shared terms “limiting”

  8. Weak transcendental base-point freeness and diameter lower bounds for the Kähler-Ricci flow

    Junsheng Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “diameter”, “lower bound”, “finite time”

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) 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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

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.