Canonical metrics on Hartogs domains

Andrea Loi; Fabio Zuddas

paper

Canonical metrics on Hartogs domains

arXiv:0805.1307

Abstract

An $n$ -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:0705.2124v1