The Third Class Of Super-cartan Domain Of The Complete Einstein-k (?) Hler Metric On Its Holomorphic Sectional Curvature

In this paper we consider the Cartan-Hartogs domain of the third typewhere RIII(q) is the third type of Cartan domain in the sense of L.K.Hua.then Z is a skew symmetric matrix of degree q . "det" denotes the determinant and Z is conjugate of Z. T denote the transpose of the matrix and q ≥ 2 is natural number.We obtain two main results: If A" we get the explicit form of complete Einstein-Kahler metric of YIII(2. q: K).At the same time we obtain the upper and lower bounds of the holomorphic sectional curvature under this metric. Yau, Cheng, Mok[1.2] proved that any bounded pseudoconvex domain Ω in Cn has a unique complete Einstein-Kahler metric.Let the metric be thethen g is a unique solution of the following Dirichlet problem of Monge-Ampere equationWhere g is called the generating function of the complete Einstein-Kahler metric of Ω. If we can obtain the explicit form of g.then the explicit form of the complete Einstein-Kahler metric is also given.We could obtain the complete Einstein-Kahler metric with explicit forms of very few pseudoconvex domains except for the bounded homogeneous domain .In this paper if K = we will prove that the generating function of the complete Einstein-Kahler metric of YIII(2,q: K) iswhere X = We obtain also the following estimate in this paperwhere w((z. w).d(z.w)) denote the holoinorpliic sectional curvature about the complete Einstein-Kahler metric of Because bounded from above by a negative constant -1.then by using the result [3] of M.Heins.we obtain the comparison theorem for the Einstein-Kahler metric and the Kobayashi metric of Comparison Theorem: If EyIII(z. w: y) and RYIIIi(z.w: y) denote the complete Einstein-Kahler metric and Kobayashi metric of Y' respectively, then there exists a positive constant C such that for all...