Paper · Oct 25, 2026

Pseudo-Calabi Flow on Toric Variety

Junhao Tian, Kai Zheng

math.DGWider flowsarXiv:2610.34340
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Abstract

In this paper, we prove the long time existence of the pseudo-Calabi flow on a compact toric variety for any -invariant initial Kähler metric. If the Mabuchi functional is proper modulo automorphisms, the flow converges exponentially to a constant scalar curvature Kähler (cscK) metric. In particular, this gives a flow proof that properness of the K-energy implies the existence of a cscK metric in the toric setting.

Comments: 55 pages

Other curvature flowsKähler geometryScalar curvatureLong-time existence & convergenceKähler–Einstein & K-stability

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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 2021, 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 Hermitian Calabi functional in complexified orbits (Jie He and Kai Zheng, 2023)7He 2023 The Ricci iteration towards cscK metrics (Kewei Zhang, 2023)3Zhang 2023 Calabi flow with bounded L^p scalar curvature (Haozhao Li, Linwei Zhang and Kai Zheng, 2023)2Li 2023 Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns (Xiaorui Yang and Hao Yu, 2025)8Yang 2025 Calabi flow with bounded L^p scalar curvature (II) (Haozhao Li and Linwei Zhang, 2025)6Li 2025 Existence of twisted Calabi flow and deformation from the J-flow to Calabi flow (Jie He and Haozhao Li, 2025)5He 2025 An improvement of regularity result for pseudo Calabi flow (Jingrui Cheng and Junhao Tian, 2026)1Cheng 2026 Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants (Haozhao Li, Bing Wang and Kai Zheng, 2026)4Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. An improvement of regularity result for pseudo Calabi flow

    Jingrui Cheng and Junhao TianWider flows PDF

    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows · Same author Junhao Tian · Shared terms “pseudo Calabi”, “Calabi flow”, “cscK metric”

  2. Calabi flow with bounded scalar curvature

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    Topics Kähler–Einstein & K-stability, Scalar curvature, Long-time existence & convergence · Same author Kai Zheng · Shared terms “Calabi flow”, “converges exponentially”

  3. The Ricci iteration towards cscK metrics

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    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows · Same author Kai Zheng · Shared terms “Calabi flow”

  5. Existence of twisted Calabi flow and deformation from the -flow to Calabi flow

    Jie He and Haozhao LiWider flows PDF

    Topics Kähler–Einstein & K-stability, Scalar curvature, Long-time existence & convergence · Shared terms “Calabi flow”, “cscK metric”, “constant scalar”

  6. Calabi flow with bounded scalar curvature (II)

    Haozhao Li and Linwei ZhangWider flows PDF

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

  7. Hermitian Calabi functional in complexified orbits

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  8. Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns

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    Topics Kähler–Einstein & K-stability, Long-time existence & convergence, Other curvature flows · Shared terms “Calabi flow”

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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. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

  1. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.