Finite-time singularities of Calabi flow

Finite-time singularities of Calabi flow concern Chen's conjecture that the Calabi flow, the fourth-order gradient flow of the Calabi energy in a Kähler class, exists for all positive times from every smooth Kähler initial metric. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), dated 24 September 2026, claims a counterexample on complex projective ten-space in the Fubini–Study class. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper or code is not presented here as independent verification.

Background

As recounted in the introduction of the manuscript:
- Calabi introduced the flow to seek canonical metrics in a Kähler class. The manuscript cites Chen's smooth long-time existence conjecture in the form of He and Zeng's Conjecture 1.1, and notes that Chen and Cheng emphasized the question for classes containing a constant-scalar-curvature Kähler metric.
- On compact Riemann surfaces, Chruściel proved global existence and convergence; Chen gave another proof, and Struwe a unified treatment of Calabi and normalized Ricci flows on surfaces. In higher dimensions, Chen and He proved local existence and stability near a constant-scalar-curvature metric, and He and Zeng developed short-time theory for rougher data.
- Continuation criteria include a uniform Ricci bound (Chen–He), a uniform scalar-curvature bound (Chen–Cheng), and, as stated by Li, Zhang and Zheng, a uniform spatial L^p scalar-curvature bound with p greater than the complex dimension. Weak Calabi flows exist globally (Streets) and converge in the d₁ distance when the class contains a constant-scalar-curvature metric (Berman–Darvas–Lu); the manuscript notes that this does not give smooth continuation.

The original result

Claim. Theorem 1.1 of A finite-time singularity of Calabi flow on projective space states that there is a smooth U(10)-invariant Kähler form on CP¹⁰, in the Fubini–Study class, whose maximal smooth Calabi flow exists on [0, T_*) with 0 < T_* < ∞, and a point p with S(ω(t))(p) ~ a(T_* − t)^−1/2 for some a > 0. The manuscript states that this disproves Chen's conjecture, even in a class containing a constant-scalar-curvature metric. Corollary 1.2 carries the singularity to products with a stationary constant-scalar-curvature factor, giving examples in every complex dimension n ≥ 10.
Method, as described: Calabi's radial ansatz in the Hwang–Singer momentum formulation reduces the flow to a scalar fourth-order equation. An exact self-similar shrinking solution of its flat limit is built from a profile obtained by matching an analytic solution at the origin to a half-line solution; a finite exact-arithmetic certificate with analytic enclosures verifies the matching inequalities. A right inverse of the flat linearized equation on all negative times and a short-time outer solution then correct the profile on the compact manifold.
Source. The manuscript A finite-time singularity of Calabi flow on projective space at commit 1581d4c, with the certificate scripts in its verification folder.
Formalization. The collection lists no Lean formalization for this result.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 19 October 2026)
2026-10-12	KitaKen1/calabi-small-energy-singularity (GitHub, commit `516d623`), manuscript Small-energy finite-time singularities of Calabi flow (Draft 1, 12 October 2026)	Extension (proposed construction)	Proposes smooth U(7)-invariant Calabi flows on CP⁷ in the Fubini–Study class whose initial potentials are arbitrarily small in the uniform norm, whose Calabi energies stay arbitrarily small (more generally, any L^p norm of the scalar curvature minus its average with 2 ≤ p < 7) and whose Sobolev constants obey one common bound, yet which develop a finite-time singularity with S ~ a(T_* − t)^−1/2 at a point. The draft states that the critical L⁷ norm diverges logarithmically, so the example does not decide continuation under a uniform L⁷ bound. The seven-dimensional profile is certified by exact rational arithmetic; the interval-arithmetic script states that it is "adapted from OpenAI release family 352 (commit fd4aeeb…)".	GitHub repository. Its README presents "a proposed proof", records that the finite arithmetic checks were rerun on 12 October 2026, and states that the repository contains no Lean formalization (its description mentions a planned one).
2026-10-16	J. Jardine, Lattice24, referee review rank_140 in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	Reports no gap. The reviewer states that it read every section, checked the geometric reduction and the profile equations by hand and with computer algebra, and reran the three certificate scripts with assertions enabled. Its one minor note is that fixed-scale existence for the outer Cauchy problem and the regularity of distributional solutions are taken from cited results (Huang 2026, Hörmander 1961, Dong–Zhang 2015) that it did not check; it states that this is not a gap.	Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0.

Notes

H. Li, B. Wang and K. Zheng, Long-time existence of the Calabi flow on Kähler surfaces with bounded Sobolev constants (arXiv:2610.05217, version 1, 4 October 2026), state that on compact Kähler surfaces the flow extends past any finite time if its Sobolev constants stay uniformly bounded, and that it exists for all time in certain classes when the initial Calabi energy is below an explicit threshold. The abstract does not mention the manuscript.
The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.