Paper · May 31, 2018

Longtime existence of Kähler Ricci flow and holomorphic sectional curvature

Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam, Freid Tong

math.DGarXiv:1805.12328
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Abstract

In this work, we obtain a existence criteria for the longtime Kähler Ricci flow solution. Using the existence result, we generalize a result by Wu-Yau on the existence of Kähler Einstein metric to the case with possibly unbounded curvature. Moreover, the Kähler Einstein metric with negative scalar curvture must be unique up to scaling.

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Submitted
May 31, 2018
Subjects
math.DG

Comments: 25 pages

Ricci flowKähler–Ricci flowEinstein metricsKähler 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 Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach (Ryosuke Nomura, 2016)4Nomura 2016 On existence and curvature estimates of Ricci flow (Man-Chun Lee and Luen-Fai Tam, 2017)8Lee 2017 Chern-Ricci flows on noncompact complex manifolds (Man-Chun Lee and Luen-Fai Tam, 2017)6Lee 2017 The Kähler-Ricci flow on manifolds with negative holomorphic curvature (Freid Tong, 2018)1Tong 2018 Instantaneously complete Chern-Ricci flow and Kähler-Einstein metrics (Shaochuang Huang, Man-Chun Lee and Luen-Fai Tam, 2019)3Huang 2019 Kähler manifolds with almost non-negative curvature (Man-Chun Lee and Luen-Fai Tam, 2019)5Lee 2019 Complete Kähler-Einstein metric on Stein manifolds with negative curvature (Man-Chun Lee, 2019)2Lee 2019 Gromov-Hausdorff limits of immortal Kähler-Ricci flows (Man-Chun Lee, Valentino Tosatti and Junsheng Zhang, 2026)7Lee 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The Kähler-Ricci flow on manifolds with negative holomorphic curvature

    Freid Tong

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Same author Freid Tong · Shared terms “holomorphic sectional”, “negative scalar”, “sectional curvature”

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

    Man-Chun Lee

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

  3. Instantaneously complete Chern-Ricci flow and Kähler-Einstein metrics

    Shaochuang Huang, Man-Chun Lee and Luen-Fai Tam

    Topics Kähler–Einstein & K-stability, Einstein metrics, Long-time existence & convergence · Same authors Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam · Shared terms “possibly unbounded”, “unbounded curvature”

  4. Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach

    Ryosuke Nomura

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “holomorphic sectional”, “sectional curvature”, “Yau”

  5. Kähler manifolds with almost non-negative curvature

    Man-Chun Lee and Luen-Fai Tam

    Topics Kähler–Ricci flow, Long-time existence & convergence, Kähler geometry · Same authors Man-Chun Lee, Luen-Fai Tam

  6. Chern-Ricci flows on noncompact complex manifolds

    Man-Chun Lee and Luen-Fai Tam

    Topics Kähler–Ricci flow, Kähler geometry · Same authors Man-Chun Lee, Luen-Fai Tam · Shared terms “existence criteria”, “Yau”

  7. Gromov-Hausdorff limits of immortal Kähler-Ricci flows

    Man-Chun Lee, Valentino Tosatti and Junsheng Zhang

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Same author Man-Chun Lee

  8. On existence and curvature estimates of Ricci flow

    Man-Chun Lee and Luen-Fai Tam

    Topics Kähler–Ricci flow, Kähler geometry · Same authors Man-Chun Lee, Luen-Fai Tam · Shared terms “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) 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. 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.