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 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 $(\prod _{j=1}^k\Omega _j)^{{\mathbb {B}}^{d_0}}(\mu )$ is defined as the Hartogs type domain constructed over the product $\prod _{j=1}^k\Omega _j$ of irreducible bounded symmetric domains $\Omega _j$ $(1\le j \le k)$, with the fiber over each point $(z_1,\ldots ,z_k)\in \prod _{j=1}^k\Omega _j$ being a ball in $\mathbb {C}^{d_0}$ of the radius $\prod _{j=1}^kN_{\Omega _j}(z_j,\overline{z_j})^{\frac{\mu _j}{2}}$ of the product of positive powers of their generic norms. Any such domain $(\prod _{j=1}^k\Omega _j)^{{\mathbb {B}}^{d_0}}(\mu )$ $(k\ge 2)$ is a bounded nonhomogeneous domain. The purpose of this paper was to obtain necessary and sufficient conditions for the metric $\alpha g(\mu )$ $(\alpha >0)$ on the domain $(\prod _{j=1}^k\Omega _j)^{{\mathbb {B}}^{d_0}}(\mu )$ 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.