Canonical metrics on some complex domains of $\C^n$

The study of the existence and uniqueness of a preferred Kaehler metric on a given complex manifold $M$ is a very important area of research. In this talk we recall the main results and open questions for the most important canonical metrics (Einstein, constant scalar curvature, extremal, Kaehler-Ricci solitons) in the compact and the non-compact case, then we consider a particular class of complex domains $D$ in $\C^n$, the so-called Hartogs domains, which can be equipped with a natural Kaehler
metric $g$. We show that if $g$ is an extremal Kaehler metric, then $(D, g)$ is holomorphically isometric to an open subset of the $n$-dimensional complex hyperbolic space. We also prove the same assertion under the assumption that there exists a real holomorphic vector field $X$ on $D$ such that $(g, X)$ is a Kaehler-Ricci soliton.