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 $\Omega\subset{\mathbb C}^{n}$ corresponds to a sub-level set of a smooth, real-valued function $\Psi$, such that the form $\omega = {\bf i}\partial\bar{\partial}\Psi$ is KÃ¤hler and has bounded curvature outside a bounded subset, then this domain admits a hermitian metric of strictly negative holomorphic sectional curvature. Keywords:Kobayashi-hyperbolicity, KÃ¤hler metric, plurisubharmonic functionCategories:32Q45, 32Q35