Journal of the
Korean Mathematical Society JKMS

In this paper we describe the Einstein-K\"ahler metric for the Cartan-Hartogs of the first type which is the special case of the Hua domains. Firstly, we reduce the Monge-Amp\` ere equation for the metric to an ordinary differential equation in the auxiliary function $X=X(z,w)=|w|^2[\det(I-ZZ^T)]^{-\frac{1}{K}}$(see below). This differential equation can be solved to give an implicit function in $X$. Secondly, we get the estimate of the holomorphic section curvature under the complete Einstein-K\"ahler metric on this domain.

Keywords: Cartan-Hartogs domain, Einstein-Kahler metric, comparison theorem, Hua domain

MSC numbers: 32H02