Yau's question on bounded holomorphic functions: does a complete simply connected Kahler manifold with sectional curvature at most $-a^2<0$ carry a nonconstant bounded holomorphic function?

Let $M$ be a complete simply connected Kahler manifold of complex dimension at least two whose real sectional curvature satisfies $\mathrm{Sec}_g\le -a^2<0$ , with no lower curvature bound assumed. A biholomorphism to a bounded domain would give bounded coordinates; the weaker question asks only for one nonconstant bounded holomorphic function. Must $M$ admit a nonconstant bounded holomorphic function, that is, is $H^\infty(M)\ne\mathbb C$ ?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Kahler geometry; several complex variables
Posed by: S.-T. Yau, Problem 38 of his 1982 problem list, as recorded by Damin Wu and S.-T. Yau (2020, p. 103; 2019 survey, Section 3)
Year posed: 1982
Years open: 44y
Solved: 2026-09-25
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 33 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for some finite $m\ge2$ there is a domain $\mathcal T\subset\mathbb C^m$ , diffeomorphic to $\mathbb R^{2m}$ , with a complete Kahler metric $g$ satisfying $\mathrm{Sec}_g\le -1$ and $H^\infty(\mathcal T)=\mathbb C$ : every bounded holomorphic function is constant, so the Caratheodory pseudometric vanishes. This disproves the universal assertion in the one-sided question, and also answers Wu's 1967 bounded-domain question negatively under its hypotheses. The sectional curvatures are unbounded below, so the question under a two-sided bound $-b^2\le\mathrm{Sec}\le -a^2$ remains open; the dimension is large and not made explicit.

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. It is the companion of the negatively pinched threefold manuscript of the same date; the release presents them as one family, and they answer different posed questions.

Verification

No independent mathematician has checked this yet. Checked here: the introduction and Theorem 1.1 were read against the one-sided question as Wu and Yau record it. lean/docs/359.md says the formalization covers only the negatively pinched companion and that this one-sided construction is separate, so there is no Lean statement for this entry. The dimension $m$ is not specified: it is some sufficiently large fixed finite value. The curvature is unbounded below, so the two-sided bounded-function question is left open, as the paper says.

Yau's question on bounded holomorphic functions: does a complete simply connected Kahler manifold with sectional curvature at most $-a^2<0$ carry a nonconstant bounded holomorphic function?

What was actually shown

What the AI did

Verification

Sources

Discussion

Yau's question on bounded holomorphic functions: does a complete simply connected Kahler manifold with sectional curvature at most −a2<0-a^2<0−a2<0 carry a nonconstant bounded holomorphic function?

What was actually shown

What the AI did

Verification

Sources

Discussion

Yau's question on bounded holomorphic functions: does a complete simply connected Kahler manifold with sectional curvature at most $-a^2<0$ carry a nonconstant bounded holomorphic function?