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.
released 2026-09-25 | 2 theorems · 21 lemmas · 27 proofs · 13,272 words |
PLAY LEVEL 1 »(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.
released 2026-09-25 | 1 theorem · 19 lemmas · 27 proofs · 24,996 words |
PLAY LEVEL 2 »(pdf)
We construct, in some sufficiently large fixed finite complex dimension m, a domain in ℂm 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.