Paper · Mar 19, 2026

An improvement of regularity result for pseudo Calabi flow

Jingrui Cheng, Junhao Tian

math.DGWider flowsarXiv:2603.19405
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Abstract

In this paper, we observe that if the initial data of pseudo Calabi flow has volume form close to a smooth one, then the flow is immediately smooth for . As an application, we show that if the initial data has volume form close to that of a cscK metric, then the pseudo Calabi flow exists for . We also prove similar improvement of regularity and long time existence result for pseudo Calabi flow on a Fano manifold when the volume form is bounded and the class is close to .

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

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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 The Ricci iteration towards cscK metrics (Kewei Zhang, 2023)6Zhang 2023 Calabi flow with bounded L^p scalar curvature (Haozhao Li, Linwei Zhang and Kai Zheng, 2023)5Li 2023 Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns (Xiaorui Yang and Hao Yu, 2025)7Yang 2025 Calabi flow with bounded L^p scalar curvature (II) (Haozhao Li and Linwei Zhang, 2025)3Li 2025 Existence of twisted Calabi flow and deformation from the J-flow to Calabi flow (Jie He and Haozhao Li, 2025)2He 2025 Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants (Haozhao Li, Bing Wang and Kai Zheng, 2026)4Li 2026 On global Chern-Calabi flows on compact complex surfaces (Li Ma and Li Sheng, 2026)8Ma 2026 Pseudo-Calabi Flow on Toric Variety (Junhao Tian and Kai Zheng, 2026)1Tian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Pseudo-Calabi Flow on Toric Variety

    Junhao Tian and Kai ZhengWider 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. Existence of twisted Calabi flow and deformation from the -flow to Calabi flow

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

  3. Calabi flow with bounded scalar curvature (II)

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  5. Calabi flow with bounded scalar curvature

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

  6. The Ricci iteration towards cscK metrics

    Kewei ZhangWider flows PDF

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

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

    Xiaorui Yang and Hao YuWider flows PDF

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

  8. On global Chern-Calabi flows on compact complex surfaces

    Li Ma and Li ShengWider flows PDF

    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) 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) 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)