Balanced metrics on the Fock-Bargmann-Hartogs domains

The Fock-Bargmann-Hartogs domain () in is defined by the inequality where , which is an unbounded non-hyperbolic domain in . This paper introduces a Kahler metric on , where is the Kahler metric associated with the Kahler potential () on . 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 with the weight for . Secondly, using the explicit expression of the Bergman kernel, we obtain the necessary and sufficient condition for the metric on the domain to be a balanced metric. So, we obtain the existence of balanced metrics for a class of Fock-Bargmann-Hartogs domains.