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 w = i∂∂ψ is Kähler and has bounded curvature outside a bounded subset, then this domain admits a hermitian metric of strictly negative holomorphic sectional curvature.