Balanced metrics on some Hartogs type domains over bounded symmetric domains

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The definition of balanced metrics was originally given by Donaldson in the case of a compact polarized Kähler manifold in 2001, who also established the existence of such metrics on any compact projective Kähler manifold with constant scalar curvature. Currently, the only noncompact manifolds on which balanced metrics are known to exist are homogeneous domains. The generalized Cartan-Hartogs domain (∏ⱼ₌₁ᵏΩⱼ)ᵇᵈ⁰(μ) is defined as the Hartogs type domain constructed over the product ∏ⱼ₌₁ᵏΩⱼ of irreducible bounded symmetric domains Ωⱼ (1≤ j ≤ k), with the fiber over each point (z₁,...,zₖ)∈ ∏ⱼ₌₁ᵏΩⱼ being a ball in Cᵈ⁰ of the radius ∏ⱼ₌₁ᵏNΩⱼ(zⱼ,zⱼ)μʲ/² of the product of positive powers of their generic norms. Any such domain (∏ⱼ₌₁ᵏΩⱼ)ᵇᵈ⁰(μ) (k≥ 2) is a bounded nonhomogeneous domain. The purpose of this paper is to obtain necessary and sufficient conditions for the metric αg(μ) (α>0) on the domain (∏ⱼ₌₁ᵏΩⱼ)ᵇᵈ⁰(μ) to be a balanced metric, where g(μ) is its canonical metric. As the main contribution of this paper, we obtain the existence of balanced metrics for a class of such bounded nonhomogeneous domains.