Wu's bounded-domain question for negatively pinched Kahler manifolds: is a complete simply connected Kahler manifold with pinched negative curvature biholomorphic to a bounded domain?

In complex dimension one, a complete simply connected surface with curvature bounded above by a negative constant is biholomorphic to the disc. H. Wu (1967) asked whether a complete simply connected Kahler manifold with nonpositive sectional curvature and holomorphic sectional curvature bounded above by a negative constant must be biholomorphic to a bounded domain. Wu and Yau (2019, Conjecture 4.3) state the two-sided version: if a simply connected complete Kahler manifold $M$ of complex dimension $n$ has every real sectional curvature between two negative constants, $-B\le K\le -A<0$ , must $M$ be biholomorphic to a bounded (pseudoconvex) domain in $\mathbb C^n$ ?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Kahler geometry; several complex variables
Posed by: H. Wu (1967, normal families paper, p. 195, question (1)); two-sided pinched form by Damin Wu and S.-T. Yau (2019 survey, Conjecture 4.3)
Year posed: 1967
Years open: 59y
Solved: 2026-09-25
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Lean-checked, statement unaudited
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 36 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: there is a contractible domain $M\subset\mathbb C^3$ (a Hartogs-type disc bundle over the unit ball of $\mathbb C^2$ ) with a smooth complete Kahler metric whose real sectional curvatures satisfy $-B\le K\le -A<0$ , such that no bounded holomorphic map $M\to\mathbb C^3$ has nowhere-vanishing Jacobian; hence $M$ is not biholomorphic to any bounded domain. This disproves Wu-Yau Conjecture 4.3 and gives a negative answer under the hypotheses of Wu's 1967 question. It does not show absence of bounded holomorphic functions (two base coordinates are bounded), and the example is Stein. Only complex dimension three is constructed.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with one fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscript is authored 'OpenAI' and names no human author. A companion manuscript of the same date (the one-sided Liouville construction) answers a different question and is listed as its own entry; the manuscript says the two constructions and proofs are independent.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against Wu's question and Wu-Yau Conjecture 4.3. formalization.yaml lists ComparatorChallenges/PinchedKahler.json with declaration OAI.PinchedHartogs.main_theorem in OAI/Geometry/Kahler/Main.lean. The challenge statement was read here: there exist an open, nonempty, contractible set $M\subset\mathbb C^2\times\mathbb C$ and a smooth Kahler metric on it that is geodesically complete with all real sectional curvatures in $[-B,-A]$ , $0<A\le B$ , such that no bounded holomorphic map to $\mathbb C^3$ has nowhere-vanishing Jacobian and $M$ is not biholomorphic to a bounded domain. That is the headline claim. Not rebuilt here. The domain is Stein and carries two bounded nonconstant holomorphic functions; the obstruction is to bounded coordinates only.

Wu's bounded-domain question for negatively pinched Kahler manifolds: is a complete simply connected Kahler manifold with pinched negative curvature biholomorphic to a bounded domain?

What was actually shown

What the AI did

Verification

Sources

Discussion