Canonical metrics on Cartan--Hartogs domains

Abstract

In this paper we address two problems concerning a family of domains \(M_{\Omega}(\mu) \subset \C^n\), called Cartan-Hartogs domains, endowed with a natural Kaehler metric \(g(\mu)\). The first one is determining when the metric \(g(\mu)\) is extremal (in the sense of Calabi), while the second one studies when the coefficient \(a_2\) in the Engli\v{s} expansion of Rawnsley \(\epsilon\)-function associated to \(g(\mu)\) is constant.