Balanced metrics on the Fock-Bargmann-Hartogs domains

The Fock-Bargmann-Hartogs domain $D_{n,m}(\mu)$ ($\mu>0$) in $\mathbb{C}^{n+m}$ is defined by the inequality $\|w\|^2<e^{-\mu\|z\|^2},$ where $(z,w)\in \mathbb{C}^n\times \mathbb{C}^m$, which is an unbounded non-hyperbolic domain in $\mathbb{C}^{n+m}$. This paper introduces a K\"{a}hler metric $\alpha g(\mu;\nu)$ $(\alpha>0)$ on $D_{n,m}(\mu)$, where $g(\mu;\nu)$ is the K\"{a}hler metric associated with the K\"{a}hler potential $\Phi(z,w):=\mu\nu{\Vert z\Vert}^{2}-\ln(e^{-\mu{\Vert z\Vert}^{2}}-\Vert w\Vert^2)$ ($\nu>-1$) on $D_{n,m}(\mu)$. The purpose of this paper is twofold. Firstly, we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space of square integrable holomorphic functions on $(D_{n,m}(\mu), g(\mu;\nu))$ with the weight $\exp\{-\alpha \Phi\}$ for $\alpha>0$. Secondly, using the explicit expression of the Bergman kernel, we obtain the necessary and sufficient condition for the metric $\alpha g(\mu;\nu)$ $(\alpha>0)$ on the domain $D_{n,m}(\mu)$ to be a balanced metric. So we obtain the existence of balanced metrics for a class of Fock-Bargmann-Hartogs domains.