Paper · Nov 1, 2026

Chern scalar curvature and finite-time singularities of the conformal Chern–Calabi flow

Hongyu Chen, Li Ma, Li Sheng

math.APWider flowsarXiv:2611.01623
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As on arXiv

Abstract

On every unit-volume flat complex three-torus, we construct a solution of the volume-preserving conformal Chern–Calabi flow with a finite-time Chern Type-I singularity at any prescribed point. The full spatial supremum has an exact asymptotic coefficient, and prescribed rescalings converge to a complete nonflat radial shrinking solution. The positive part of Chern scalar curvature can remain arbitrarily small throughout the flow, as can each prescribed subcritical spatial curvature norm. We determine the punctured terminal metric, its one-point completion and unique nonflat tangent cone, and the terminal energy asymptotics. We also exclude Chern Type-II under a logarithmically improved critical weak bound on Chern scalar curvature, and prove a separate scale-invariant integral criterion involving first Chern–Ricci curvature. The constructed singular solutions satisfy both criteria.

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Submitted
Nov 1, 2026
Subjects
math.AP, math.DG
Other curvature flowsSingularitiesKähler geometryScalar curvatureKähler–Einstein & K-stability

Its neighbourhood

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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.

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Continuation and convergence for the higher dimensional conformal Chern–Calabi flow

    Hongyu Chen, Li Ma and Li ShengWider flows

    Topics Kähler–Einstein & K-stability, Scalar curvature, Other curvature flows · Same authors Hongyu Chen, Li Ma, Li Sheng · Shared terms “conformal Chern”, “Chern scalar”, “Calabi flow”

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

    Li Ma and Li ShengWider flows

    Topics Kähler–Einstein & K-stability, Other curvature flows, Kähler geometry · Same authors Li Ma, Li Sheng · Shared terms “Chern”, “Calabi flow”

  3. Calabi flow with bounded scalar curvature (II)

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

  4. Calabi flow with bounded scalar curvature

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

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

    Jie He and Haozhao LiWider flows

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

  6. Pseudo-Calabi Flow on Toric Variety

    Junhao Tian and Kai ZhengWider flows

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

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

    Haozhao Li, Bing Wang and Kai ZhengWider flows

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

  8. Calabi flow on polarized Kähler surfaces

    Haozhao Li, Bing Wang and Kai ZhengWider flows

    Topics Kähler–Einstein & K-stability, Scalar curvature, Other curvature flows · 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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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)

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.

  3. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.