Paper · Dec 14, 2017

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

Xiuxiong Chen, Tamás Darvas, Weiyong He

math.DGWider flowsarXiv:1712.05095
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As on arXiv

Abstract

We prove a compactness theorem for Kähler metrics with various bounds on Ricci curvature and the functional. We explore applications of our result to the continuity method and the Calabi flow.

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Submitted
Dec 14, 2017
Subjects
math.DG, math.AP, math.CV
Journal
Calc. Var. PDE 58 (2019), no. 4, Paper No. 139
Other curvature flowsKähler geometryLimit spaces & RCDKähler–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 Comparison of the Calabi and Mabuchi geometries and applications to geometric flows (Tamás Darvas, 2016)1Darvas 2016 Remarks of weak-compactness along Kahler Ricci flow (Xiuxiong Chen and Bing Wang, 2016)4Chen 2016 On the extension and smoothing of the Calabi flow on complex tori (Hongnian Huang, 2016)7Huang 2016 The Calabi flow with rough initial data (Weiyong He and Yu Zeng, 2017)3He 2017 Improved explicit estimates for the discrete Laplace operator with hyperbolic circle patterns (Aijin Lin and Longxiang Wu, 2025)8Lin 2025 Calabi flow with bounded L^p scalar curvature (II) (Haozhao Li and Linwei Zhang, 2025)5Li 2025 Calabi flow on polarized Kähler surfaces (Haozhao Li, Bing Wang and Kai Zheng, 2026)2Li 2026 Convergence of the Calabi flow on Kähler surfaces with bounded Sobolev constants (Haozhao Li, Bing Wang and Kai Zheng, 2026)6Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Comparison of the Calabi and Mabuchi geometries and applications to geometric flows

    Tamás Darvas

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Same author Tamás Darvas · Shared terms “Kähler metrics”, “Calabi”

  2. Calabi flow on polarized Kähler surfaces

    Haozhao Li, Bing Wang and Kai ZhengWider flows

    Topics Kähler–Einstein & K-stability, Limit spaces & RCD, Other curvature flows · Shared terms “Calabi flow”, “compactness theorem”, “continuity”

  3. The Calabi flow with rough initial data

    Weiyong He and Yu ZengWider flows

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Same author Weiyong He · Shared terms “Calabi flow”

  4. Remarks of weak-compactness along Kahler Ricci flow

    Xiuxiong Chen and Bing Wang

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

  5. Calabi flow with bounded scalar curvature (II)

    Haozhao Li and Linwei ZhangWider flows

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Shared terms “Calabi flow”

  6. Convergence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

    Haozhao Li, Bing Wang and Kai ZhengWider flows

    Topics Kähler–Einstein & K-stability, Limit spaces & RCD, Other curvature flows · Shared terms “Calabi flow”

  7. On the extension and smoothing of the Calabi flow on complex tori

    Hongnian HuangWider flows

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Shared terms “Calabi flow”

  8. Improved explicit estimates for the discrete Laplace operator with hyperbolic circle patterns

    Aijin Lin and Longxiang WuWider flows

    Topics Kähler–Einstein & K-stability, Limit spaces & RCD, Other curvature flows · Shared terms “Calabi flow”, “compactness”

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.

Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) 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) Kähler–Einstein & K-stability A Fano manifold carries a Kähler–Einstein metric exactly when it is K-polystable, an algebraic condition (Chen–Donaldson–Sun, Tian). Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

  1. 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.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

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

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.