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.