Balanced metrics on the Fock-Bargmann-Hartogs domains

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The Fock-Bargmann-Hartogs domain Dₙ,ₘ(μ) (μ>0) in Cⁿ⁺ᵐ is defined by the inequality w²<e⁻μᶻ², where (z,w)∈ Cⁿ× Cᵐ, which is an unbounded non-hyperbolic domain in Cⁿ⁺ᵐ. This paper introduces a Kähler metric αg(μ;ν) (α>0) on Dₙ,ₘ(μ), where g(μ;ν) is the Kähler metric associated with the Kähler potential Φ(z,w):=μν z²-(e-μ z²- w²) (ν>-1) on Dₙ,ₘ(μ). 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 (Dₙ,ₘ(μ), g(μ;ν)) with the weight {-αΦ} for α>0. Secondly, using the explicit expression of the Bergman kernel, we obtain the necessary and sufficient condition for the metric αg(μ;ν) (α>0) on the domain Dₙ,ₘ(μ) to be a balanced metric. So we obtain the existence of balanced metrics for a class of Fock-Bargmann-Hartogs domains.