The metric compactification of a Kobayashi hyperbolic complex manifold and a Denjoy--Wolff Theorem

We study the metric compactification of a Kobayashi hyperbolic complex manifold $\mathcal{X}$ equipped with the Kobayashi distance ${\mathsf{k}}_{\mathcal{X}}$. We show that this compactification is genuine – i.e., $\mathcal{X}$ embeds as a dense open subset – even without completeness of ${\mathsf{k}}_{\mathcal{X}}$, and that it becomes a \emph{good compactification} in the sense of Bharali–Zimmer when $(\mathcal{X},{\mathsf{k}}_{\mathcal{X}})$ is complete. As an application, we obtain a criterion for the continuous extension of quasi-isometric embeddings from $(\mathcal{X},{\mathsf{k}}_{\mathcal{X}})$ into visibility domains of complex manifolds. For a Kobayashi hyperbolic domain $\mathrm{\Omega }⊊\mathcal{X}$, to each boundary point of $\mathrm{\Omega }$ in the end compactification, we associate a fiber of metric boundary points. This allows the small and big horospheres of Abate to be expressed as the intersection and union of horoballs centered at metric boundary points. We use this to formulate a Wolff-type lemma in terms of horoballs and prove a Denjoy–Wolff theorem for complete hyperbolic domains satisfying a boundary divergence condition for ${\mathsf{k}}_{\mathcal{X}}$. Finally, we present necessary and sufficient conditions under which the identity map extends continuously between the metric and end compactifications.