On canonical metrics on Cartan-Hartogs domains

The Cartan-Hartogs domains are defined as a class of Hartogs type domains over irreducible bounded symmetric domains. The purpose of this paper is twofold. Firstly, for a Cartan-Hartogs domain endowed with the canonical metric we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space of square integrable holomorphic functions on with the weight (where is a globally defined Kahler potential for ) for , and, furthermore, we give an explicit expression of the Rawnsley's -function expansion for Secondly, using the explicit expression of the Rawnsley's -function expansion, we show that the coefficient of the Rawnsley's -function expansion for the Cartan-Hartogs domain is constant on if and only if is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.