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 $Ω^{B^{d_0}}(μ)$ endowed with the canonical metric $g(μ)$, we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space $\mathcal{H}_α$ of square integrable holomorphic functions on $(Ω^{B^{d_0}}(μ), g(μ))$ with the weight $\exp\{-αφ\}$ (where $φ$ is a globally defined Kähler potential for $g(μ)$) for $α>0$, and, furthermore, we give an explicit expression of the Rawnsley's $\varepsilon$-function expansion for $(Ω^{B^{d_0}}(μ), g(μ)).$ 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 $(Ω^{B^{d_0}}(μ), g(μ))$ is constant on $Ω^{B^{d_0}}(μ)$ if and only if $(Ω^{B^{d_0}}(μ), g(μ))$ is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.