The Fourth Category Hua Structure Of The Einstein-k (?) Hler Metric

In this paper, we consider the fourth Hua construction which is introduced by Yin Weiping. Letwhere R_IV(n) is the fourth Cartan domain in the sense of Hua, (Z|-)^t indicates the conjugate and transpose of Z, n is a positive integer number. K and p are positive real numbers.Mok and Yau have proved that any bounded pseudo-convex domain in Cⁿ has a unique complete Einstein-Kahler metric Wu pointed that among the four classical invariant metrics ,the Einstein-Kahler metric is the hardest to compute because its existence is proved by complicated nonconstructive methods. It is very difficult to write down the metric except on homogeneous domains ,where the metric is determined by the invariant volume form which is unique up to a multiplicative constant.The Bergman kernel functions on the fourth Hua construction HC_IV are obtained in explicit formulas ,and we know the Bergman kernel of the domains are exhaustive . so HC_IV is a bounded pseudo-convex domain, thus it has a unique complete Einstein-Kahler metric. In this paper, we reduce the higher dimensional non-linear complex Monge-Ampere equation for the metric to an ordinary differential equation in the auxiliary X = X(Z, ζ, η). We give the generating function of the Einstein-Kahler metric on HC_IV.Then we give the explicit forms of the complete Einstein-Kahler metric when in some special cases of K and p. In these cases, we get the holomorphic sectional curvature of the Einstein-Kahler metric.