The Hartogs Domain Of The Cartan-einstein-k (?) Hler Metric

In this paper we discuss the Einstein-Kahler metric on the super-Cartan domain of the third type. Firstly we get the implicit solution of the generating function of the Einstein-Kahler metric on the super-Cartan domain of the third type. Secondly we obtain the holomorphic sectional curvature under the Einstein-Kahler metric and get its estimation, hence we get the comparison theorem for the complete Einstein-Kahler metric and the Kobayashi metric on the domain YIII. Thirdly we get the complete Einstein-Kahler metric with explicit form on a non-homogeneity domain. According to the paper [8], we know that it is very difficult to get the explicit form of a complete Einstein-Kahler metric .In this paper we adopt a special method by using the holomorphic automorphism group of the super-Cartan domain of the third type and the invariant founction X under the holomorphic automorphism to reduce the Monge-Ampere into the ordinary differential equation. Cheng and Yau[l]have proved that there exists an unique complete Einstein-Kahler metric on the bounded pseudoconvex domain in Cn with continuous twice differentiate boundary. Then N. Mok and S. T. Yau [2] extend the result to any bounded pseudoconvex domain . Assuming the complete Einstein-Kahler metric of isthen g is the unique solution of the following Dirichlet problem of the Monge-Ampere equation:Here g is called the generating function of the Einstein-Kahler measure on . The definition of the super-Cartan domain of the third type iswhere Rm(g) is the Cartan domain of the third type in the sense of Hua , Z is the conjugation of Z, the superior letter T is the transpose of a matrix, det is the determinant of a matrix, q > 2 is a natural number. The super-Cartan domain is the special case of the Hua domains.The Bergman kernel function of Ym is given in papers papers[4][5]. Hence this Bergman kernel function is exhaustion, so YIII(q)is a pseudoconvex domain. By using the holomorphic automorphism group of the super-Cartan domain of the third type and the invariant founction X under the holomorphic automorphism we reduce the Monge-Ampere equation into the following ordinary differential equation problem:This problem has the following implicit solution:And we obtain the the generating function g of the Einstein-Kahler measure on as follows:here Y is the solution of the above problem and also is a function of X ,Y =dY/dX. We also obtain the holomorphic sectional curvature under the Einstein-Kahler metric of YIII as follows:Where , the holomorphic sectional curvature under the Einstein-Kahler metric of YIII is bounded from above by a negative constant, then we get the comparison theorem for the complete Einstein-Kahler metric and the Kobayashi metric on YIII.Comparison theorem: When be the complete Einstein-Kahler metric and the Kobayashi metric of YIII respectively, then there exists positive constant c such thatis held for any (z,w) YIII,. When K = ,the generating function of the Einstein-Kahler metric on YIII isThen we get the explicit form of the complete Einstein-Kahler metric on YIII, were YIII is the non-homogeneous domain in general. It is the first time that the complete Einstein-Kahler metric with explicit form is obtained on the non-homogeneous domain.