COMPLETE EINSTEIN-KÄHLER METRIC AND HOLOMORPHIC SECTIONAL CURVATURE ON YII(r, p;K)

The explicit complete Einstein-Kähler metric on the second type Cartan-Hartogs domain YII(r, p;K) is obtained in this paper when the parameter K equals p 2 + 1 p+1 . The estimate of holomorphic sectional curvature under this metric is also given which intervenes between −2K and − 2K p and it is a sharp estimate. In the meantime we also prove that the complete Einstein-Kähler metric is equivalent to the Bergman metric on YII(r, p;K) when K = p 2 + 1 p+1 . Introduction It is well known that the Bergman, Carathéodory, Kobayashi and EinsteinKähler metrics are four classical invariant metrics in complex analysis. The Bergman metric was introduced by S.Bergman for one variable in 1921 and for several variables in 1933.C.Carathéodory introduced the invariant distance in 1926 and H.Reiffen introduced the invariant metric in 1963, therefore the Carathéodory metric is also called Carathéodory-Reiffen metric. The Kobayashi metric was introduced by S.Kobayashi in 1967 and by H.Royden in 1970. Therefore the Kobayashi metric is also called Kobayashi-Royden metric. 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 of this metric is defined to be −∂∂logdet(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, it requires the metric to be complete.According to a famous article of Wu, one knows that Einstein-Kähler metric is the most difficult to compute among the four metrics because its existence is proved by complicate nonconstructive methods. If we normalize the metric by requiring the scalar curvature to be minus one, then the Einstein-Kähler metric is unique. Cheng and Yau proved that any bounded pseudo convex domainD with continuous second partial derivatives boundary admits a complete Einstein-Kähler metric. Mok and Yau have extended this result to an arbitrary bounded pseudo convex domain in C. If this Einstein-Kähler metric is given by ED(z) := ∑ ∂g ∂zi∂zj dzidzj , Date: 10th June 2005. 2000 Mathematics Subject Classification. 32H15, 32F07, 32F15.