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 $\mathrm{\parallel }w{\mathrm{\parallel }}^2<e^{-\mu \mathrm{\parallel }z{\mathrm{\parallel }}^2},$ 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 $(D_{n,m}(\mu), g(\mu;\nu))$withtheweight$\exp{-\alpha \Phi}$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.