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The Fock–Bargmann–Hartogs domain  () in  is defined by the inequality  where , which is an unbounded non-hyperbolic domain in . This paper introduces a Kähler metric   on , where  is the Kähler metric associated with the Kähler 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.

Balanced metrics on the Fock–Bargmann–Hartogs domains