Canonical metrics on some domains of ${ℂ}^n$

Zuddas, Fabio

Séminaire de théorie spectrale et géométrie, Tome 27 (2008-2009), pp. 143-156.

The study of the existence and uniqueness of a preferred Kähler 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, Kähler-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 a Kähler-Einstein, constant scalar curvature, extremal or a soliton metric then $(D,g)$ is holomorphically isometric to an open subset of the $n$-dimensional complex hyperbolic space. If $D$ is bounded, we also show the same assertion under the assumption that $g$ is a scalar multiple of the Bergman metric.

The results we present are proved in papers joint with A. Loi and A. J. Di Scala ([11], [20]).

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