Monge-amp¨¨re Equation In A Class Of Hartogs Domains, Complete Einstein-k (?) Hler Metric

Invariant metrics are intensively studied in the field of Several Complex Variables.Traditionally, three metrics, namely, Bergman, Carath(?)odary and Kobayashi are counted as classical. A popular recent addition is the Einstein-K(a|¨)hler metric, so there are four classical invariant metrics nowadays. Due to the study of the Calabi conjecture, we know that on a K(a|¨)hler manifold, the existence of Einstein-K(a|¨)hler metric can be reduced to the existence of solutions of the Monge-Amp(?)re equations. Furthermore, the existence of a complete Einstein-K(a|¨)hler metric on any bounded pseudoconvex domain was proved by Mok and Yau.The Bergman metric on a homogeneous domain is Einstein-Kahler, as the Ricci curvature of it is a negative constant. However, for general domains, this is not the case. It is usually hard to find an explicit formula of the Einstein-K(a|¨)hler metric on a non-homogeneous bounded pseudoconvex domain, for the approach taken by Mok and Yau is not constructive. Since an explicit form of Einstein-K(a|¨)hler metric is helpful to the study of boundary invariants and asymptotic behavior of domain, computing an explicit form of complete Einstein-K(a|¨)hler metric on common bounded domain is of great significance.In this paper, we study a type of Hartogs domain called Cartan-Hartogs domain such as (?) = {(ξ,z,ω)∈C×I_m1,n1×I_m2,n2 : |ξ|²ï¼œÏ†^p(z,z)iψ^q(ω,ω)}. Its base space is a direct product of two first type Cartan domains, which means the base space of this Hartogs domain is reducible while the domain is irreducible. As for domain (?), we obtain three results: the implicit function form of the generating function of the Einstein-K(a|¨)hler metric on this domain with slight suppositions; the explicit form of Einstein-K(a|¨)hler metric on this domain when the parameters are in certain conditions; when the parameters have certain values, we get a Reinhardt domain and we also give the explicit form of the complete Einstein-K(a|¨)hler metric on this Reinhardt domain. As for the non-homogeneous domain, it is the first time to get a non-homogeneous Reinhardt domain .