Remarks on the canonical metrics on the Cartan-Hartogs domains

Abstract

The Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. For a Cartan-Hartogs domain ${\Omega }^B(\mu )$ endowed with the natural Kähler metric $g(\mu ),$ Zedda conjectured that the coefficient $a_2$ of the Rawnsley's $\epsilon$-function expansion for the Cartan-Hartogs domain $({\Omega }^B(\mu ),g(\mu ))$ is constant on ${\Omega }^B(\mu )$ if and only if $({\Omega }^B(\mu ),g(\mu ))$ is biholomorphically isometric to the complex hyperbolic space. In this paper, following Zedda's argument, we give a geometric proof of the Zedda's conjecture by computing the curvature tensors of the Cartan-Hartogs domain $({\Omega }^B(\mu ),g(\mu ))$.

11 pages, to appear in Comptes Rendus Mathematique