Paper · May 4, 2016

Remarks of weak-compactness along Kahler Ricci flow

Xiuxiong Chen, Bing Wang

math.DGarXiv:1605.01374
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Abstract

We provide a direct proof of time-slice weak compactness along the Kähler Ricci flow on Fano manifolds.

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Submitted
May 4, 2016
Subjects
math.DG

Comments: 5 figures

Ricci flowKähler–Ricci flowKähler geometryLimit spaces & RCD

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 Fano bundles (Xin Fu and Shijin Zhang, 2016)7Fu 2016 The local entropy along Ricci flow—Part A: the no-local-collapsing theorems (Bing Wang, 2017)4Wang 2017 On a twisted conical Kähler-Ricci flow (Yashan Zhang, 2017)5Zhang 2017 Compactness of Kähler metrics with bounds on Ricci curvature and I functional (Xiuxiong Chen, Tamás Darvas and Weiyong He, 2017)3Chen 2017 Kahler-Ricci flow on blowups along submanifolds (Bin Guo, 2018)6Guo 2018 Kähler-Ricci flow on mathbf G-spherical Fano manifolds (Feng Wang and Xiaohua Zhu, 2023)8Wang 2023 Geometric regularity of blow-up limits of the Kähler-Ricci flow (Max Hallgren, Wangjian Jian, Jian Song et al., 2023)2Hallgren 2023 Two Comments on "Laplace comparison on Kahler Ricci flow and convergence (arXiv:2509.14820v1)" (Xiuxiong Chen and Bing Wang, 2025)1Chen 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Two Comments on "Laplace comparison on Kahler Ricci flow and convergence (arXiv:2509.14820v1)"

    Xiuxiong Chen and Bing Wang

    Topics Kähler–Ricci flow, Kähler geometry · Same authors Xiuxiong Chen, Bing Wang

  2. Geometric regularity of blow-up limits of the Kähler-Ricci flow

    Max Hallgren, Wangjian Jian, Jian Song et al.

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Shared terms “time slice”

  3. Compactness of Kähler metrics with bounds on Ricci curvature and I functional

    Xiuxiong Chen, Tamás Darvas and Weiyong HeWider flows

    Topics Limit spaces & RCD, Kähler geometry · Same author Xiuxiong Chen · Shared terms “compactness”

  4. The local entropy along Ricci flow—Part A: the no-local-collapsing theorems

    Bing Wang

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same author Bing Wang

  5. On a twisted conical Kähler-Ricci flow

    Yashan Zhang

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry

  6. Kahler-Ricci flow on blowups along submanifolds

    Bin Guo

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry

  7. The Kähler-Ricci flow on Fano bundles

    Xin Fu and Shijin Zhang

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Shared terms “Fano”

  8. Kähler-Ricci flow on -spherical Fano manifolds

    Feng Wang and Xiaohua Zhu

    Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Shared terms “Fano manifolds”

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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

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. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  3. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.