Einstein-K(a)hler metric on Cartan-Hartogs domain of the second type

The Einstein-Kahler metric for the Cartan-Hartogs domain of the second type is described. Firstly, the Monge-Ampère 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.     

The comparison theorem for Bergman and Kobayashi metrics on Cartan-Hartogs domain of the second type

In this paper, the holomorphic sectional curvature under invariant metric on a Cartan-Hartogs domain of the second type YII(N,p,K) is presented and an invariant K?]lher metric which is complete and not less than the Bergman metric is constructed, such that its holomorphic sectional curvature is bounded above by a negative constant. Hence a comparison theorem for the Bergman and Kobayashi metrics on YII(N,p,K) is obtained.     

Extremal problems on the generalized Hua domain of the first type

Some extremal problems between the generalized Hua domain of the first type and the unit ball are studied. The extremal mapping and extremal value in explicit formulas are also obtained.