A Cartan-the Hartogs Domain Of The K (?) Hler-einstein Metrics

In this paper, we consider Cartan-Hartogs domain of the following type which is introduced by Yin Weiping and Roos.Where R_I(m,n) is the first Cartan domain in the sense of Loo-Keng HUA, det indicates the determinate, Z indicates the conjugate and transpose of Z, r,m, and n are positive integer numbers, K is a positive real number. || · || is the standardHermitian norm in C^r, i.e. η = (η₁,η₂,......, η_r) ∈C, then ||η||² = .Cheng-Yau and Mok-Yau have showed that any bounded pseudo-convex domain in C^N has a unique complete Kahler-Einstein metric. The Bergman kernal functions on Cartan-Hartogs domains are obtained in explicit formulas by Yin Weiping, so we can easily prove that Y_I is a bounded pseudo-convex domain. It is very difficult to write down the complete Kahler-Einstein metric with explicit forms of very few pseudo-convex domains except on bounded homogeneous domains.In this paper, by using the automorphism subgroup of Y_I and the bi-holomorphic invariance, we reduce the higher dimensional non-linear complex Monge-Ampere equation for the metric to an ordinary differential equation by using some special skill, we obtain the generating function of Y_I and its explicit forms Kahler-Einstein metric on Cartan-Hartogs domain, and we give the explicit forms of the complete Kahler-Einstein metric when in some special cases of K, the holomorphic sectional curvature of the invariant Kahler-Einstein metric on it. In this cases, we get the upper and lower bounds of the holomorphic sectional curvature, and we obtain the comparison theorem of complete Kahler-Einstein metric and Kobayashi metric.