2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/160658An $n$-dimensional Hartogs domain $D_F$ with strongly pseudoconvex boundary can be equipped with a natural Kaehler metric $g_F$. This paper contains two results. In the first one we prove that if $g_F$ is an extremal Kaehler metric then $(D_F, g_F)$ is holomorphically isometric to an open subset of the $n$-dimensional complex hyperbolic space. In the second one we prove the same assertion under the assumption that there exists a real holomorphic vector field $X$ on $D_F$ such that $(g_F, X)$ is a Kaehler-Ricci soliton.This paper contains the results about extremal metrics described in arXiv:0705.2124v1Differential GeometryComplex Variables53C55; 32Q15; 32T15text