On canonical metrics on Cartan-Hartogs domains

Abstract: 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 $\Omega^{B^{d_0}}(\mu)$ endowed with the canonical metric $g(\mu)$, we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space $\mathcal{H}_{\alpha}$ of square integrable holomorphic functions on $(\Omega^{B^{d_0}}(\mu), g(\mu))$ with the weight $\exp{-\alpha \varphi}$ (where $\varphi$ is a globally defined K\"{a}hler potential for $g(\mu)$) for $\alpha>0$, and, furthermore, we give an explicit expression of the Rawnsley's $\varepsilon$-function expansion for $(\Omega^{B^{d_0}}(\mu), g(\mu)).$ Secondly, using the explicit expression of the Rawnsley's $\varepsilon$-function expansion, we show that the coefficient $a_2$ of the Rawnsley's $\varepsilon$-function expansion for the Cartan-Hartogs domain $(\Omega^{B^{d_0}}(\mu), g(\mu))$ is constant on $\Omega^{B^{d_0}}(\mu)$ if and only if $(\Omega^{B^{d_0}}(\mu), g(\mu))$ is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.