Paper · Mar 21, 2018

The Kähler-Ricci flow on pseudoconvex domains

Huabin Ge, Aijin Lin, Liangming Shen

math.DGarXiv:1803.07761v2
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Abstract

We establish the existence of Kähler-Ricci flow on pseudoconvex domains with general initial metric without curvature bounds. Moreover we prove that this flow is simultaneously complete, and its normalized version converge to the complete Kähler-Einstein metric, which generalizes Topping's works on surfaces.

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Submitted
Mar 21, 2018
Last revised
Mar 26, 2018 (v2)
Subjects
math.DG

Comments: 17 pages, Theorem 1.3 is strengthened and combined to Theorem 1.2

Ricci flowKähler–Ricci flowEinstein metricsSurfacesKähler geometryKähler–Einstein & K-stability

Its neighbourhood

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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)8Chau 2017 The Kähler-Ricci flow on manifolds with negative holomorphic curvature (Freid Tong, 2018)5Tong 2018 Longtime existence of Kähler Ricci flow and holomorphic sectional curvature (Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al., 2018)6Huang 2018 The K"ahler-Ricci flow with Log Canonical Singularities (Albert Chau, Huabin Ge, Ka-Fai Li et al., 2019)4Chau 2019 Complete Kähler-Einstein metric on Stein manifolds with negative curvature (Man-Chun Lee, 2019)3Lee 2019 On the Kahler-Ricci flow on Fano manifolds (Bin Guo, Duong H. Phong and Jacob Sturm, 2020)7Guo 2020 Fiberwise Kähler-Ricci flows on families of bounded strongly pseudoconvex domains (Young-Jun Choi and Sungmin Yoo, 2020)2Choi 2020 The Kähler-Ricci soliton on bounded pseudoconvex domains (Zehao Sha, 2024)1Sha 2024
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  1. The Kähler-Ricci soliton on bounded pseudoconvex domains

    Zehao Sha

    Topics Kähler–Einstein & K-stability, Einstein metrics, Kähler geometry · Shared terms “pseudoconvex domain”

  2. Fiberwise Kähler-Ricci flows on families of bounded strongly pseudoconvex domains

    Young-Jun Choi and Sungmin Yoo

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

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

    Man-Chun Lee

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

  4. The K"ahler-Ricci flow with Log Canonical Singularities

    Albert Chau, Huabin Ge, Ka-Fai Li et al.

    Topics Einstein metrics · Same authors Huabin Ge, Liangming Shen

  5. 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”

  6. 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

  7. On the Kahler-Ricci flow on Fano manifolds

    Bin Guo, Duong H. Phong and Jacob Sturm

    Topics Kähler–Einstein & K-stability, Kähler–Ricci flow, Einstein metrics

  8. 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 Liangming Shen · Shared terms “initial metric”

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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

  1. 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.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

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