The Einstein-kähler Metric with Explicit Formulas on Some Non-homogeneous Domains

In this paper we describe the Einstein-Kähler metric for the Cartan-Hartogs domains which are the special case of the Hua domains. First of all, we reduce the Monge-Ampère equation for the metric to an ordinary differential equation in the auxiliary function X = X(z, w). This differential equation can be solved to give an implicit function in X. Secondly, for some cases, we obtained explicit forms of the complete Einstein-Kähler metrics on Cartan-Hartogs domains which are the non-homogeneous domains. Let M be a complex manifold. Then a Hermitian metric ∑ i,j gi,jdz i ⊗ dz defined on M is said to be Kähler if the Kähler form Ω = √−1∑i,j gi,jdz ∧ dz is closed. The Ricci form is given by −∂∂ log det(gi,j). If the Ricci form of the Kähler metric is proportional to the Kähler form, the metric is called Einstein-Kähler. If the manifold is not compact, we require the metric to be complete. Clearly for a noncompact complex manifold to admit such a metric, it is necessary that there exists a volume form, the negative of whose Ricci tensor defines a complete Kähler metric. The volume form of this Kähler metric must be equivalent to the original volume form. If we normalize the metric by requiring the scalar curvature to be minus one, then the Einstein-Kähler metric is unique. Cheng and Yau[CY] proved that any bounded domain D which is the intersection of domain with C boundary admits a complete Einstein-Kähler. Without any regularity assumption on the domain D, Mok and Yau[MY] proved that the complete Einstein-Kähler metric always exists. This Einstein-Kähler metric is given by ED(z) := ∑ ∂g ∂zi∂zj dzidzj , where g is an unique solution to the boundary problem of the Monge-Ampère equation: ⎧⎨ ⎩ det ( ∂g ∂zi∂zj ) = e z ∈ D, g = ∞ z ∈ ∂D, We call g the generating function of ED(z). It is obvious that if one determines g explicitly, then the Einstein-Kähler metric is also determined explicitly. Therefore if one would like to compute the Einstein-Kähler metric explicitly, it suffices to compute the generating function g in explicit formula. The explicit formulas for the Einstein-Kähler metric, however, are only known on homogeneous domains. In his famous paper [Wu], H.Wu points out that among the four classical invariant metrics(i.e. the Bergman metric, Carathéodory metric, ∗Received April 3, 2003; accepted for publication September 8, 2003. Research supported in part by NSF of China (Grant No. 10171051 and 10171068) and NSF of Beijing (Grant No. 1002004 and 1012004). †Department of Mathematics, Capital Normal University, Beijing 100037, P. R. China ([email protected]). ‡ [email protected] or [email protected]