Paper · Nov 21, 2018

A note on compact -solutions of Kähler-Ricci flow

Yuxing Deng, Xiaohua Zhu

math.DGarXiv:1811.08859
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Abstract

In this paper, we give a complete classification of -solutions of Kähaler-Ricci flow on compact complex manifolds. Namely, they must be quotients of products of irreducible compact Hermitian symmetric manifolds.

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Submitted
Nov 21, 2018
Subjects
math.DG

Comments: 7 pages

Ricci flowKähler–Ricci flowKähler geometry

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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 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 Asymptotic behavior of positively curved steady Ricci Solitons, II (Yuxing Deng and Xiaohua Zhu, 2016)3Deng 2016 Singular limits of Kähler-Ricci flow on Fano G-manifolds (Yan Li, Gang Tian and Xiaohua Zhu, 2018)4Li 2018 Ancient solutions to the Kähler Ricci flow (Yu Li, 2020)8Li 2020 Uniformly strong convergence of Kähler-Ricci flows on a Fano manifold (Feng Wang and Xiaohua Zhu, 2020)1Wang 2020 Horosymmetric limits of Kähler-Ricci flow on Fano G-manifolds (Gang Tian and Xiaohua Zhu, 2022)7Tian 2022 Kähler-Ricci flow on mathbf G-spherical Fano manifolds (Feng Wang and Xiaohua Zhu, 2023)5Wang 2023 Laplace comparison on Kähler Ricci flow and convergence (Gang Tian, Qi S. Zhang, Zhenlei Zhang et al., 2025)2Tian 2025 Weighted volume comparison and monotonicity for L^p-bound of Bakry-Émery Ricci curvature (Jintao Ye and Xiaohua Zhu, 2026)6Ye 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Uniformly strong convergence of Kähler-Ricci flows on a Fano manifold

    Feng Wang and Xiaohua Zhu

    Topics Kähler–Ricci flow, Kähler geometry · Same author Xiaohua Zhu · Shared terms “compact complex”, “complex manifold”

  2. Laplace comparison on Kähler Ricci flow and convergence

    Gang Tian, Qi S. Zhang, Zhenlei Zhang et al.

    Topics Kähler–Ricci flow, Kähler geometry · Same author Xiaohua Zhu

  3. Asymptotic behavior of positively curved steady Ricci Solitons, II

    Yuxing Deng and Xiaohua Zhu

    Topics Kähler geometry · Same authors Yuxing Deng, Xiaohua Zhu

  4. Singular limits of Kähler-Ricci flow on Fano -manifolds

    Yan Li, Gang Tian and Xiaohua Zhu

    Topics Kähler–Ricci flow, Kähler geometry · Same author Xiaohua Zhu

  5. Kähler-Ricci flow on -spherical Fano manifolds

    Feng Wang and Xiaohua Zhu

    Topics Kähler–Ricci flow, Kähler geometry · Same author Xiaohua Zhu

  6. Weighted volume comparison and monotonicity for -bound of Bakry-Émery Ricci curvature

    Jintao Ye and Xiaohua Zhu

    Topics Kähler–Ricci flow, Kähler geometry · Same author Xiaohua Zhu

  7. Horosymmetric limits of Kähler-Ricci flow on Fano -manifolds

    Gang Tian and Xiaohua Zhu

    Topics Kähler–Ricci flow, Kähler geometry · Same author Xiaohua Zhu

  8. Ancient solutions to the Kähler Ricci flow

    Yu Li

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “complete classification”

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) 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.