Family 359Differential geometryLeanNot yet digested
Negative Kähler curvature without bounded holomorphic coordinates
Constructs a contractible domain in ℂ 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.
Manuscripts in this family · 2
A negatively pinched Kähler threefold without bounded holomorphic coordinates
Sep 25, 2026·paper.pdfLean
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 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
Sep 25, 2026·paper.pdf
We construct, in some sufficiently large fixed finite complex dimension m, a domain in ℂ diffeomorphic to 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.