Negative Kähler curvature without bounded holomorphic coordinates

What OpenAI claims

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)

Papers (2)

1. A negatively pinched Kähler threefold without bounded holomorphic coordinates

   25 September 2026 · Preprint, not peer reviewed · PDF

   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.

2. One-sided negative sectional curvature and the holomorphic Liouville property

   25 September 2026 · Preprint, not peer reviewed · PDF

   We construct, in some sufficiently large fixed finite complex dimension m, a domain in ℂm diffeomorphic to 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.

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.