Paper · Oct 16, 2025

Calabi flow with bounded scalar curvature (II)

Haozhao Li, Linwei Zhang

math.DGWider flowsarXiv:2510.14302v2
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Abstract

In this paper, we show that on a compact Kähler manifold the Calabi flow can be extended as long as some space-time integrals of the scalar curvature are bounded.

Comments: 34 pages, an appendix added

Other curvature flowsKähler geometryScalar curvatureKähler–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 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 Calabi flow with bounded L^p scalar curvature (Haozhao Li, Linwei Zhang and Kai Zheng, 2023)1Li 2023 Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns (Xiaorui Yang and Hao Yu, 2025)7Yang 2025 Twisted Calabi functional and twisted Calabi flow (Jie He and Haozhao Li, 2025)4He 2025 Existence of twisted Calabi flow and deformation from the J-flow to Calabi flow (Jie He and Haozhao Li, 2025)2He 2025 An improvement of regularity result for pseudo Calabi flow (Jingrui Cheng and Junhao Tian, 2026)6Cheng 2026 The Calabi flow with prescribed curvature on finite graphs (Yi Li, Jie Wang, Pingsan Yuan et al., 2026)8Li 2026 Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants (Haozhao Li, Bing Wang and Kai Zheng, 2026)3Li 2026 Pseudo-Calabi Flow on Toric Variety (Junhao Tian and Kai Zheng, 2026)5Tian 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Calabi flow with bounded scalar curvature

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  2. 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, Other curvature flows · Same author Haozhao Li · Shared terms “Calabi flow”, “compact Kähler”

  3. Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants

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  4. Twisted Calabi functional and twisted Calabi flow

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

  5. Pseudo-Calabi Flow on Toric Variety

    Junhao Tian and Kai ZhengWider flows PDF

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

  6. An improvement of regularity result for pseudo Calabi flow

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

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

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

  8. The Calabi flow with prescribed curvature on finite graphs

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

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. 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) 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.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.