Negative Kähler curvature without bounded holomorphic coordinates
OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
OpenAI's result summary
Constructs a contractible domain in ℂ^(3) 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.
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MANUSCRIPT 01
A negatively pinched Kähler threefold without bounded holomorphic coordinates
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 ℂ^(3) with nowhere-vanishing Jacobian and is therefore not biholomorphic to a bounded domain. This gives a negative answer…
Full source abstract
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 ℂ^(3) 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.
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One-sided negative sectional curvature and the holomorphic Liouville property
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…
Full source abstract
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.
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