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 $\big(\prod_{ j=1}^k\Omega_j\big)^{ { \mathbb{ B}}^{ d_0}}(\mu)$ is defined as the Hartogs type domain constructed over the product $\prod_{ j=1}^k\Omega_j$ of irreducible bounded symmetric domains $\Omega_j$ $(1\leq j \leq k)$, with the fiber over each point $(z_1,...,z_k)\in \prod_{ j=1}^k\Omega_j$ being a ball in $\mathbb{ C}^{ d_0}$ of the radius $\prod_{ j=1}^kN_{ \Omega_j}(z_j,\bar{ z_j})^{ \frac{ \mu_j}{ 2}}$ of the product of positive powers of their generic norms. Any such domain $\big(\prod_{ j=1}^k\Omega_j\big)^{ { \mathbb{ B}}^{ d_0}}(\mu)$ $(k\geq 2)$ 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 $\big(\prod_{ j=1}^k\Omega_j\big)^{ { \mathbb{ B}}^{ d_0}}(\mu)$ 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.
Categories: math.CV math.DG