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 $\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right)$ with the weight $\exp \{-\alpha \varphi \}$ (where $\varphi$ is a globally defined Kähler potential for $g(\mu )$) for $\alpha >0$, and, furthermore, we give an explicit expression of the Rawnsley’s $\varepsilon$-function expansion for $\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right) .$ 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 $\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right)$ is constant on $\Omega ^{B^{d_0}}(\mu )$ if and only if $\left( \Omega ^{B^{d_0}}(\mu ), g(\mu )\right)$ is biholomorphically isometric to the complex hyperbolic space. So we give an affirmative answer to a conjecture raised by M. Zedda.