Balanced metrics on the Fock–Bargmann–Hartogs domains

Abstract

The Fock–Bargmann–Hartogs domain $D_{n,m}(\mu )$ ($\mu >0$) in $\mathbb {C}^{n+m}$ is defined by the inequality $\Vert w\Vert ^2<e^{-\mu \Vert z\Vert ^2},$ 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 Kähler metric $\alpha g(\mu ;\nu )$ $(\alpha >0)$ on $D_{n,m}(\mu )$, where $g(\mu ;\nu )$ is the Kähler metric associated with the Kähler 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.