Negative Kähler curvature without bounded holomorphic coordinates

Constructs a contractible domain in ℂ³ with a complete negatively pinched Kähler metric but no bounded holomorphic coordinates, disproving bounded-domain uniformization in this setting. A higher-dimensional example has sectional curvature at most −1 and only constant bounded holomorphic functions; its curvature is not bounded below. (Lean)

Manuscripts (2)

A negatively pinched Kähler threefold without bounded holomorphic coordinates

September 25, 2026 · source · Lean formalization available · cite as \cite{OAI:A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026}

We construct a contractible domain in complex dimension three with a complete Kähler metric whose real sectional curvatures lie between two finite negative constants. It admits no bounded holomorphic map to ℂ³ with nowhere-vanishing Jacobian and is therefore not biholomorphic to a bounded domain. This gives a negative answer to the negatively pinched Kähler uniformization question.

One-sided negative sectional curvature and the holomorphic Liouville property

September 25, 2026 · source · cite as \cite{OAI:One-sided-negative-sectional-curvature-and-the-holomorphic-Liouville-property-September-25-2026}

We construct, in some sufficiently large fixed finite complex dimension m, a domain in ℂ^m diffeomorphic to \mathbb R^{2m} that admits a complete Kähler metric with real sectional curvature at most −1 and has only constant bounded holomorphic functions. Its sectional curvatures are unbounded below. The construction gives a negative answer to the one-sided bounded-holomorphic-function question.

Result 359 · Differential geometry. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.