Yau's uniformization conjecture
Yau's uniformization conjecture asks whether a complete noncompact Kähler manifold with positive holomorphic bisectional curvature must be biholomorphic to complex Euclidean space; Yau posed it in his 1982 problem survey. A manuscript in OpenAI's public mathematics collection (github.com/openai/math), dated 23 September 2026, claims the conjecture in every dimension under pointwise strict positivity, with no further curvature, volume-growth or topological hypothesis. The family has no Lean formalization. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
Background
- As recounted in the introduction of the manuscript:
- In dimension one the statement follows from Cohn-Vossen's theorem and the parabolicity theorem of Blanc, Fiala and Huber. In the compact case, positive bisectional curvature characterizes complex projective space, by the work of Mori and of Siu and Yau on the Frankel conjecture.
- Earlier results add hypotheses: Greene and Wu proved Steinness under positive sectional curvature; Mok obtained an affine-algebraic embedding under maximal volume growth and quadratic curvature decay; Chau and Tam proved uniformization under bounded nonnegative bisectional curvature and maximal volume growth; Liu, and later Lee and Tam, removed the upper curvature bound while keeping maximal volume growth. Chen and Zhu, Ni and Tam, and Cao's Harnack estimates are further cited antecedents.
- Recent work treats special cases: Datar, Pingali and Seshadri prove that complete noncompact Kähler surfaces of positive sectional curvature are biholomorphic to C2, and give a criterion under positive bisectional curvature; Wu (arXiv:2609.18506) weakens the topological hypotheses of that criterion; Dat (arXiv:2609.18171) develops an m-Hessian approach in dimensions at least three. The manuscript notes that several of these papers record AI assistance.
The original result
- Claim. Theorem 1.1 of Uniformization of complete Kähler manifolds with positive bisectional curvature states that a connected noncompact complex manifold of dimension n ≥ 1 that admits a smooth complete Kähler metric with strictly positive holomorphic bisectional curvature is biholomorphic to Cn. The positivity is pointwise, with no uniform lower bound, no upper curvature bound, and no volume-growth, noncollapsing, Ricci-pinching or topological assumption.
- Method, as described: a global Kähler–Ricci flow is built from complete Hermitian approximations on exhausting domains, and scalar comparison is proved before completeness of the flow is known. A minimization of the boundary dual norm over holomorphic discs in the cotangent bundle gives a viscosity inequality and hence matrix and scalar Harnack inequalities. Shrinking holomorphic charts are then assembled by complex dynamics, with quantitative control of the coordinate changes, to obtain a biholomorphism onto all of Cn rather than onto a proper domain.
- Source. The manuscript Uniformization of complete Kähler manifolds with positive bisectional curvature at commit
1581d4c. - Formalization. The repository has no Lean scope note for this family.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 20 October 2026) |
|---|---|---|---|---|
| 2026-10-16 | J. Jardine, Lattice24, referee review 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 all sections and checked the flow, heat-correction and comparison steps by hand. Its one minor note concerns the nested limits in the disc-deformation lemma and the disc inequality: the reviewer checked the bookkeeping structurally but not line by line, and names the uniformity of a Taylor remainder along the minimizing sequence as the main risk, which the paper addresses explicitly. | 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
- The earlier uniformization results are due to the authors named in the Background section, as recounted by the manuscript.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.