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 K"{a}hler manifold in 2001, who also established the existence of such metrics on any compact projective K"{a}hler 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 $({\prod }_{j=1}^k{\mathrm{\Omega }}_j{)}^{{\mathbb{B}}^{d_0}}(\mu )$ is defined as the Hartogs type domain constructed over the product ${\prod }_{j=1}^k{\mathrm{\Omega }}_j$ of irreducible bounded symmetric domains ${\mathrm{\Omega }}_j$ $(1\le j\le k)$, with the fiber over each point $(z_1,\mathrm{.}\mathrm{.}\mathrm{.},z_k)\in {\prod }_{j=1}^k{\mathrm{\Omega }}_j$ being a ball in ${\mathbb{C}}^{d_0}$ of the radius ${\prod }_{j=1}^k{N}_{{\mathrm{\Omega }}_j}(z_j,\overline{z_j}{)}^{\frac{{\mu }_j}{2}}$ of the product of positive powers of their generic norms. Any such domain $({\prod }_{j=1}^k{\mathrm{\Omega }}_j{)}^{{\mathbb{B}}^{d_0}}(\mu )$ $(k\ge 2)$ is a bounded nonhomogeneous domain. The purpose of this paper is to obtain necessary and sufficient conditions for the metric $\alpha g(\mu )$ $(\alpha >0)$ on the domain $({\prod }_{j=1}^k{\mathrm{\Omega }}_j{)}^{{\mathbb{B}}^{d_0}}(\mu )$ to be a balanced metric, where $g(\mu )$ 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.