Paper · Oct 10, 2022

Calabi Symmetry and the Continuity Method

Hosea Wondo

math.DGarXiv:2210.04546
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Abstract

We study the convergence and curvature blow up of La Nave and Tian's continuity method on a generalised Hirzebruch surface. We show that the Gromov-Hausdorff convergence is similar to that of the Kahler-Ricci flow and obtain curvature estimates. We also show that a general solution to the continuity method either exist or all times, or the scalar curvature blows up. This behavior is known to be exhibited by the Kahler-Ricci flow.

Comments: 18 pages, 1 figures. Any comments are welcomed

Ricci flowKähler–Ricci flowSingularitiesKähler geometryScalar curvatureLimit spaces & RCD

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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 Curvature Estimates for the Continuity Method (Hosea Wondo, 2021)1Wondo 2021 Kähler-Ricci flow and conformal submersion (Hoan Nguyen, 2023)6Nguyen 2023 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)2Hallgren 2023 Finite time singularities of the Kähler-Ricci flow (Wangjian Jian, Jian Song and Gang Tian, 2023)4Jian 2023 Singularity Types for Long-Time Chern-Ricci Flow (Hosea Wondo, 2024)7Wondo 2024 On the tangent flow to the collapsing Kähler-Ricci flow on Hirzebruch surfaces (Jiangtao Li, 2025)3Li 2025 Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers (Yu Li and Junsheng Zhang, 2026)8Li 2026 Type I estimates for the Kähler-Ricci flow II (Wangjian Jian and Jian Song, 2026)5Jian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Curvature Estimates for the Continuity Method

    Hosea Wondo PDF

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

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

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

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Shared terms “Gromov Hausdorff”

  3. On the tangent flow to the collapsing Kähler-Ricci flow on Hirzebruch surfaces

    Jiangtao Li PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “Hirzebruch surface”

  4. Finite time singularities of the Kähler-Ricci flow

    Wangjian Jian, Jian Song and Gang Tian PDF

    Topics Kähler–Ricci flow, Limit spaces & RCD, Scalar curvature · Shared terms “Calabi symmetry”, “Gromov Hausdorff”

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

    Wangjian Jian and Jian Song PDF

    Topics Kähler–Ricci flow, Limit spaces & RCD, Scalar curvature · Shared terms “Hausdorff convergence”, “curvature estimates”, “Gromov Hausdorff”

  6. Kähler-Ricci flow and conformal submersion

    Hoan Nguyen PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “Calabi symmetry”, “Gromov”

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

    Hosea Wondo PDF

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Hosea Wondo

  8. Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers

    Yu Li and Junsheng Zhang PDF

    Topics Kähler–Ricci flow, Limit spaces & RCD, Scalar curvature · Shared terms “Gromov Hausdorff”, “Hausdorff convergence”

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) 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) 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)

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.