Balanced metrics on some Hartogs type domains over bounded symmetric domains

The definition of balanced metrics was originally given by Donaldson in the case of a compact polarized Kahler manifold in 2001, who also established the existence of such metrics on any compact projective Kahler 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 is defined as the Hartogs type domain constructed over the product of irreducible bounded symmetric domains , with the fiber over each point being a ball in of the radius of the product of positive powers of their generic norms. Any such domain is a bounded nonhomogeneous domain. The purpose of this paper was to obtain necessary and sufficient conditions for the metric on the domain to be a balanced metric, where 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.