Yau's uniformization conjecture
In his 1982 problem survey Yau asked whether a complete noncompact Kahler manifold with positive holomorphic bisectional curvature must be biholomorphic to , a noncompact counterpart of the Frankel conjecture (settled by Mori and Siu-Yau). It was known under extra hypotheses: maximal volume growth (Chau-Tam, Liu, Lee-Tam), curvature decay or pinching, and recently for surfaces with positive sectional curvature (Datar-Pingali-Seshadri). Is every complete connected noncompact Kahler manifold of complex dimension with strictly positive holomorphic bisectional curvature biholomorphic to ?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Kahler geometry, several complex variables
- Posed by
- S.-T. Yau, Survey on partial differential equations in differential geometry, Seminar on Differential Geometry, Ann. of Math. Stud. 102 (1982), Section 9(b), p. 45
- Year posed
- 1982
- Years open
- 44y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 58 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: if a connected noncompact complex manifold of dimension carries a smooth complete Kahler metric with pointwise strictly positive holomorphic bisectional curvature, then it is biholomorphic to ; no curvature bounds, volume growth, noncollapsing or topological hypotheses are assumed. The proof runs a Kahler-Ricci flow built from exhaustions, proves Harnack inequalities by a disc-minimization argument, obtains shrinking holomorphic charts and assembles them by polynomial-automorphism dynamics into a biholomorphism onto all of . It does not say the metric is flat, and it does not treat nonnegative curvature.
What the AI did
Produced by an unreleased internal OpenAI model as part of an OpenAI evaluation on open research problems. The release README says the vast majority of results used one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result; this result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is authored as OpenAI with no human author named. The README also cautions that unformalized results could have issues.
Verification
No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1, read against Yau's 1982 question as cited. The flow, Harnack and dynamics arguments, which are long and analytic, were not refereed. No Lean main result. The theorem uses pointwise strict positivity, which is the conjecture's hypothesis; versions with nonnegative curvature are not claimed.