On Hyperbolicity of Domains with Strictly Pseudoconvex Ends

This article establishes a sufficient condition for Kobayashi hyperbolicity of unbounded domains in terms of curvature. Specifically, when Ω ⊂ ℂn corresponds to a sub-level set of a smooth, real-valued function Ψ such that the form ω = is Kähler and has bounded curvature outside a bounded subset, then this domain admits a hermitian metric of strictly negative holomorphic sectional curvature.

HERMITIAN METRIC WITH CONSTANT HOLOMORPHIC SECTIONAL CURVATURE ON CONVEX DOMAINS

Let D be a bounded convex domain in [Formula: see text] with a Hermitian metric [Formula: see text] of constant negative holomorphic sectional curvature such that all components [Formula: see text] blow up to infinity on the boundary of D. Then D is biholomorphic to the Euclidean ball.