K\"ahler geometry of bounded pseudoconvex Hartogs domains

Let $\Omega$ be a bounded pseudoconvex Hartogs domain. There exists a natural complete K\"ahler metric $g^{\Omega}$ in terms of its defining function. In this paper, we study two problems. The first one is determining when $g^{\Omega}$ is Einstein or extremal. The second one is the existence of holomorphic isometric immersions of $(\Omega, g^{\Omega})$ into finite or infinite dimensional complex space forms.