Chen's conjecture on smooth long-time existence of the Calabi flow
Calabi introduced the Calabi flow , , to search for extremal and constant-scalar-curvature Kahler metrics in a fixed Kahler class of a compact Kahler manifold. Short-time existence and stability near cscK metrics were proved by Chen and He, and global existence is known on Riemann surfaces. Chen conjectured that the flow exists smoothly for all positive time from arbitrary smooth Kahler initial data (He-Zeng 2021, Conjecture 1.1); Chen and Cheng emphasized the case of classes containing a cscK metric. Does the smooth Calabi flow starting from any smooth Kahler metric on a compact Kahler manifold exist for all time?
- Result
- Disproved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Kahler geometry, geometric flows
- Posed by
- Xiuxiong Chen
- Year posed
- —
- Years open
- —
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 35 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: there is a smooth -invariant Kahler form on in the Fubini-Study class whose maximal smooth Calabi flow exists only on with , and at a point the scalar curvature blows up like . Since the class contains the cscK Fubini-Study metric, smooth global existence fails even there. Corollary 1.2 gives finite-time singularities in every complex dimension on . Not shown: anything in complex dimensions 2 to 9, or about weak (Streets, Berman-Darvas-Lu) flows, which continue to exist.
What the AI did
The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript (September 24, 2026). Its finite shooting step is certified by an exact integer interval-arithmetic Python program reproduced in the appendix.
Verification
No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 were read against the conjecture as the manuscript cites it; Theorem 1.1 is a counterexample to the conjecture as stated. The proof (radial momentum reduction, a self-similar shrinking profile of the flat limit, parabolic correction on the compact manifold) was not refereed, and the exact-arithmetic certificate in the appendix was not re-run here. No Lean formalization accompanies this manuscript.