Paper · Oct 7, 2019

Complete Kähler-Einstein metric on Stein manifolds with negative curvature

Man-Chun Lee

math.DGarXiv:1910.02565
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As on arXiv

Abstract

We show the existence of complete negative Kähler-Einstein metric on Stein manifolds with negatively pinched holomorphic sectional curvature. We prove that any Kähler metrics on such manifolds can be deformed to the complete negative Kähler-Einstein metric using the normalized Kähler-Ricci flow.

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Submitted
Oct 7, 2019
Subjects
math.DG, math.CV
Ricci flowKähler–Ricci flowEinstein metricsKähler geometryKähler–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 The Kähler-Ricci flow on pseudoconvex domains (Huabin Ge, Aijin Lin and Liangming Shen, 2018)5Ge 2018 The Kähler-Ricci flow on manifolds with negative holomorphic curvature (Freid Tong, 2018)2Tong 2018 Longtime existence of Kähler Ricci flow and holomorphic sectional curvature (Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al., 2018)1Huang 2018 Instantaneously complete Chern-Ricci flow and Kähler-Einstein metrics (Shaochuang Huang, Man-Chun Lee and Luen-Fai Tam, 2019)6Huang 2019 Kähler manifolds with almost non-negative curvature (Man-Chun Lee and Luen-Fai Tam, 2019)7Lee 2019 Higher-Order Estimates of Long-Time Solutions to the Kähler-Ricci Flow (Frederick Tsz-Ho Fong and Man-Chun Lee, 2020)4Fong 2020 L^∞ estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation (Wangjian Jian and Yalong Shi, 2023)8Jian 2023 Gromov-Hausdorff limits of immortal Kähler-Ricci flows (Man-Chun Lee, Valentino Tosatti and Junsheng Zhang, 2026)3Lee 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Longtime existence of Kähler Ricci flow and holomorphic sectional curvature

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

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

  2. The Kähler-Ricci flow on manifolds with negative holomorphic curvature

    Freid Tong

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “complete Kähler”, “holomorphic sectional”, “normalized Kähler”

  3. 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 · Shared terms “normalized Kähler”

  4. Higher-Order Estimates of Long-Time Solutions to the Kähler-Ricci Flow

    Frederick Tsz-Ho Fong and Man-Chun Lee

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

  5. The Kähler-Ricci flow on pseudoconvex domains

    Huabin Ge, Aijin Lin and Liangming Shen

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “complete Kähler”

  6. 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, Kähler geometry · Same author Man-Chun Lee

  7. Kähler manifolds with almost non-negative curvature

    Man-Chun Lee and Luen-Fai Tam

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

  8. estimates for Kähler-Ricci flow on Kähler-Einstein Fano manifolds: a new derivation

    Wangjian Jian and Yalong Shi

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics · Shared terms “normalized Kähler”

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.