Kähler 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.