Paper · Nov 12, 2021

Curvature Estimates for the Continuity Method

Hosea Wondo

math.DGarXiv:2111.06615v2
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Abstract

We obtain curvature estimates for long time solutions of the continuity method on compact Kähler manifolds with semi-ample canonical line bundles. In this setting, initiated in arXiv:1410.3157 and arXiv:0709.0990, we adapt arguments from arXiv:1903.05939 for the Kähler-Ricci flow to this setup. As an application, we derive curvature bounds for general metrics on product manifolds.

Comments: Corrected errors in previous version

Ricci flowKähler–Ricci flowKähler geometryLong-time existence & convergenceAnalytic minimal model program

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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 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 On almost nonpositive k-Ricci curvature (Kai Tang, 2021)4Tang 2021 On the Ricci curvature of Kahler-Ricci Flow (Cheuk Yan Fung, 2021)1Fung 2021 Calabi Symmetry and the Continuity Method (Hosea Wondo, 2022)2Wondo 2022 Independence of Singularity Type for Numerically Effective Kähler-Ricci Flows (Hosea Wondo and Zhou Zhang, 2023)3Wondo 2023 Singularity Types for Long-Time Chern-Ricci Flow (Hosea Wondo, 2024)7Wondo 2024 Higher-order Ricci estimates along immortal Kähler-Ricci flows (Wenrui Kong, 2026)8Kong 2026 Convergence of Scalar Curvature of Long Time Kähler-Ricci Flow on Kähler Manifold (Lei Zhang and Zhenlei Zhang, 2026)6Zhang 2026 On Miyaoka-Yau Inequalities and Weil-Petersson Metrics (Alexander Bednarek, 2026)5Bednarek 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. On the Ricci curvature of Kahler-Ricci Flow

    Cheuk Yan Fung PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Long-time existence & convergence · Shared terms “ample canonical”, “semi ample”, “canonical line”

  2. Calabi Symmetry and the Continuity Method

    Hosea Wondo PDF

    Topics Kähler–Ricci flow, Kähler geometry · Same author Hosea Wondo · Shared terms “continuity”, “curvature estimates”

  3. Independence of Singularity Type for Numerically Effective Kähler-Ricci Flows

    Hosea Wondo and Zhou Zhang PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Long-time existence & convergence · Same author Hosea Wondo · Shared terms “semi ample”

  4. On almost nonpositive -Ricci curvature

    Kai Tang PDF

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

  5. On Miyaoka-Yau Inequalities and Weil-Petersson Metrics

    Alexander Bednarek PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “semi ample”, “canonical line”, “ample canonical”

  6. Convergence of Scalar Curvature of Long Time Kähler-Ricci Flow on Kähler Manifold

    Lei Zhang and Zhenlei Zhang PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Kähler geometry · Shared terms “compact Kähler”, “long time”

  7. Singularity Types for Long-Time Chern-Ricci Flow

    Hosea Wondo PDF

    Topics Kähler–Ricci flow, Long-time existence & convergence, Kähler geometry · Same author Hosea Wondo

  8. Higher-order Ricci estimates along immortal Kähler-Ricci flows

    Wenrui Kong PDF

    Topics Analytic minimal model program, Kähler–Ricci flow, Long-time existence & convergence · Shared terms “curvature estimates”, “compact Kähler”, “curvature bounds”

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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) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.