Einstein-K(?)hler Metric On Cartan-Hartogs Domain Of The Second Type

It is well known that any bounded pseudoconvex domain in Cn has a complete Einstein-Kahler metric with negative Ricci curvature. Let the Einstein-Kahler metric on is then g is a solution of Monge-Ampere equation and satisfies the Dirichlet condition, and g is unique solution:where g is called generating function of the Einstein-Kahler metric on 0. We consider theSuper-Cartan domain of the second type introduced by Yin Weiping. The definition of the Super-Cartan domain of the second type is:YII(1,p; K) = {W C, Z RII(p) : |W|2K < det(I - ZZ),K >0} := YII, where RII(p) is the second type of symmetric classical domain.In this paper,the Einstein-Kahler metric for YII is described. Firstly, the Monge-Ampere equation for the metric to an ordinary differential equation in the auxiliary function X = X(z, w) is reduced by which an implicit function in X is obtained. Secondly, for some cases, the explicit forms of the complete Einstein-Kahler metrics on Cartan-Hartogs domains which are the non-homogeneous domains are obtained. Thirdly, the estimate of holomorphic sectional curvature under the Einstein-Kahler metric is given, and in some cases the comparison theorem for Kobayashi metric and Einstein-Kahler metric on Cartan-Hartogs domain of the second type is established.