Paper · Apr 3, 2019

The Kähler Ricci flow around complete bounded curvature Kähler metrics

Albert Chau, Man-Chun Lee

math.DGarXiv:1904.02233v2
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Abstract

We produce complete bounded curvature solutions to Kähler-Ricci flow with existence time estimates, assuming only that the initial data is a smooth \K metric uniformly equivalent to another complete bounded curvature \K metric. We obtain related flow results for non-smooth as well as degenerate initial conditions. We also obtain a stability result for complex space forms under the flow.

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Submitted
Apr 3, 2019
Last revised
Apr 7, 2019 (v2)
Subjects
math.DG

Comments: 22 pages; editions made to earlier version (including revision of title, addition to Theorem 1.2 and correction in proof of Claim 2.2)

Ricci flowKähler–Ricci flowSphere theoremsKähler geometryWeak & singular flowsStability & uniqueness

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 Kähler-Ricci flow of cusp singularities on quasi projective varieties (Albert Chau, Ka-Fai Li and Liangming Shen, 2017)1Chau 2017 Longtime existence of Kähler Ricci flow and holomorphic sectional curvature (Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al., 2018)6Huang 2018 Weakly PIC1 manifolds with maximal volume growth (Fei He and Man-Chun Lee, 2018)5He 2018 Complete Kähler-Einstein metric on Stein manifolds with negative curvature (Man-Chun Lee, 2019)3Lee 2019 Rigidity of area non-increasing maps (Man-Chun Lee, Luen-Fai Tam and Jingbo Wan, 2023)4Lee 2023 Expanding Ricci solitons coming out of weakly PIC1 metric cones (Pak-Yeung Chan, Man-Chun Lee and Luke T. Peachey, 2024)2Chan 2024 A note on Ricci flow from small curvature concentration and a Morrey-type condition (Albert Chau and Adam Martens, 2026)8Chau 2026 Uniqueness of Ricci flows from singular initial data (Laura Flower, Man-Chun Lee and Felix Schulze, 2026)7Flower 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Kähler-Ricci flow of cusp singularities on quasi projective varieties

    Albert Chau, Ka-Fai Li and Liangming Shen

    Topics Kähler–Ricci flow, Kähler geometry · Same author Albert Chau · Shared terms “complete bounded”, “bounded curvature”, “equivalent”

  2. Expanding Ricci solitons coming out of weakly PIC1 metric cones

    Pak-Yeung Chan, Man-Chun Lee and Luke T. Peachey

    Topics Sphere theorems, Weak & singular flows, Kähler geometry · Same author Man-Chun Lee

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

    Man-Chun Lee

    Topics Kähler–Ricci flow, Kähler geometry · Same author Man-Chun Lee · Shared terms “Kähler metrics”

  4. Rigidity of area non-increasing maps

    Man-Chun Lee, Luen-Fai Tam and Jingbo Wan

    Topics Sphere theorems, Stability & uniqueness · Same author Man-Chun Lee

  5. Weakly PIC1 manifolds with maximal volume growth

    Fei He and Man-Chun Lee

    Topics Sphere theorems · Same author Man-Chun Lee

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

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

    Topics Kähler–Ricci flow, Kähler geometry · Same author Man-Chun Lee

  7. Uniqueness of Ricci flows from singular initial data

    Laura Flower, Man-Chun Lee and Felix Schulze

    Topics Weak & singular flows, Stability & uniqueness · Same author Man-Chun Lee · Shared terms “initial data”

  8. A note on Ricci flow from small curvature concentration and a Morrey-type condition

    Albert Chau and Adam Martens

    Topics Ricci flow · Same author Albert Chau · Shared terms “complete bounded”, “bounded curvature”, “equivalent”

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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  2. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  3. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.