In this paper,we prove the vanishing of the space of square integrable harmonic(r,s)-forms relative to the Bergman metric for r+s≠N+mn on the Cartan-Hartogs domain of the first type in CN+mn.

The Cartan-Hartogs domain Y_Ⅱ(N,p;K)is the special cases of Hua domains, when K=p/2+1/(p+1),the solution of the Dirichlet problem of complex Monge-Ampère equation for domain Y_Ⅱ(N,p;K),and the explicit formula of complete K■hler-Einstein metric of Y_Ⅱ(N,p;K) is obtained.

In this paper,some extremal problems between the Cartan-Hartogs domain on k＜1 and the unit ball are studied,and the extremal mapping and extremal value in explicit formulas are obtained.

By means of holomorphic automorphic map to compute the determinant of Bergman metic matrix detT for domains Y_(II),which are Cartan-Hartogs domains of the second type,in explicit formulas and the biholomorphic invariant J_(Y_(II)) for the domains Y_(II) are obtained.

The super-Cartan domain of the second type is: YⅡ(N,p;k):={w∈CN,Z∈RⅡ(p):‖w‖2k<det(I-ZZT)}(k>0),where RⅡ(p) denotes the Cartan domains of the second type in the sence of Hua,respectively.

The purpose of this paper is to prove that every proper holomorphic self-mapping of certain Hartogs domains must be an automorphism.

It is a very important aspect in modular representationos of finite groups that how one can calculate the Cartan invariant,especially,the first Cartan invariant for a finite group of Lie type.