Canonical metrics on Hartogs domains

Abstract

An $n$-dimensional Hartogs domain $D_{F}$ can be equipped with a natural Kähler metric $g_{F}$. This paper contains two results. In the first one we prove that if $g_{F}$ is an extremal Kähler 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 Kähler--Ricci soliton.