Balanced metrics on some Hartogs type domains over bounded symmetric domains

Abstract

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.