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A finite-time singularity of Calabi flow on projective space
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Abstract
We construct a smooth Kähler metric in the Fubini–Study class of $\mathbb{CP}^{10}$ whose Calabi flow develops unbounded scalar curvature in finite time. This disproves Chen's smooth long-time existence conjecture, even in a class containing a constant-scalar-curvature metric.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Possibly relevant literature (found by the agent, not checked)
- KitaKen1, calabi-small-energy-singularity (GitHub repository, "Draft 1, 12 October 2026"; proposed as a conjecture in formal-conjectures issue #7199, 2026-10-12). A proposed proof, posted after this paper, of finite-time Calabi-flow singularities from $U(7)$-invariant metrics on $\mathbb{CP}^7$ with small Calabi energy, a uniform Sobolev constant and scalar-curvature blow-up like $(T_*-t)^{-1/2}$, the same rate as in Theorem 1.1; the README does not mention this paper.
- Haozhao Li, Bing Wang and Kai Zheng, Regularity scales and convergence of the Calabi flow (arXiv:1501.01851, 2015; Journal of Geometric Analysis 28, 2018). Its estimates for curvature blowup at possible finite-time singularities, including a scalar-curvature lower rate under a type-I hypothesis, provide context for the manuscript's claimed pointwise scalar-curvature asymptotic of order $(T_*-t)^{-1/2}$.
- Haozhao Li, Bing Wang, Kai Zheng, Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants (arXiv:2610.05217, 2026-10-04). A continuation criterion for complex surfaces (extension past any finite time when Sobolev constants stay uniformly bounded), relevant to the paper's discussion of curvature-based extension criteria; it does not mention this paper.
- Gábor Székelyhidi, The Calabi functional on a ruled surface (arXiv:math/0703562; Ann. Sci. Éc. Norm. Supér. 42 (2009), 837–856). Studies Calabi flow from metrics with momentum-construction symmetry, the same Calabi/Hwang–Singer reduction to a one-variable profile that the paper uses for its $U(10)$-invariant metrics.
- Wangjian Jian, Jian Song, Gang Tian, Finite time singularities of the Kähler-Ricci flow (arXiv:2310.07945, 2023). Shows that Kähler–Ricci flow solutions with Calabi symmetry develop Type I singularities; a second-order analogue of the $U(n)$-invariant singularity analysis in this paper.
Digests and expositions
- Latent Seminar, A finite-time Calabi-flow singularity: certified profile and compact correction (7 October 2026). This detailed exposition explains the radial reduction, certified shooting and compact correction, and reports successful arithmetic checks while explicitly distinguishing them from independent verification of the analytic arguments and full theorem. [family-level]
- Gradient Damage, Calabi Flow, Affine Bernstein Problem and 7 More in Differential Geometry (video compilation, 15 October 2026). The chapter beginning at 1:03 gives a short illustrated introduction to Calabi flow and the release's claimed result, with a link to this manuscript. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 352
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026,
author = {{OpenAI}},
title = {{A finite-time singularity of Calabi flow on projective space}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026/paper.pdf}{OAI:A-finite-time-singularity-of-Calabi-flow-on-projective-space-September-24-2026}},
year = {2026}
}From OpenAI's README for this paper:
Verification
The exact certificate and supplementary rational checks require only Python 3's
standard library. Run with assertions enabled: do not use -O or -OO, and
unset PYTHONOPTIMIZE as below. These commands use a temporary copy because the
allowances check writes arithmetic_allowances.json beside its source file.
(
set -eu
work_dir="$(mktemp -d)"
trap 'rm -rf "$work_dir"' 0 HUP INT TERM
cp -R verification "$work_dir/verification"
unset PYTHONOPTIMIZE
python3 -B "$work_dir/verification/certificate.py"
python3 -B "$work_dir/verification/arithmetic_allowances.py"
# Optional supplementary root and tail check:
python3 -B "$work_dir/verification/tail-check.py"
)
The certificate is silent on success. The allowances check prints a report with
all_exact_rational_checks_pass set to true, and the optional tail check prints
a message that all checks passed exactly. Successful execution means the
programs' assertions hold; it does not independently verify their analytic
interpretation or the full theorem.
Family 352: A finite-time singularity of Calabi flow
Disproves Chen's smooth long-time existence conjecture for Calabi flow by constructing a smooth $U(10)$-invariant Kähler metric on $\mathbb{CP}^{10}$ whose flow develops a finite-time singularity. The metric lies in the Fubini–Study class, so the failure occurs even in a class containing a constant-scalar-curvature metric.
This is the only paper in this family.