Complete Einstein-Kähler Metric and Holomorphic Sectional Curvature on $Y_{II}(r,p;K)$

Abstract

The explicit complete Einstein-Kähler metric on the second type Cartan-Hartogs domain $Y_{II}(r,p;K)$ is obtained in this paper when the parameter $K$ equals $\frac{p}{2}+\frac{1}{p+1}$. The estimate of holomorphic sectional curvature under this metric is also given which intervenes between $-2K$ and $-\frac{2K}{p}$ 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 $Y_{II}(r,p;K)$ when $K=\frac{p}{2}+\frac{1}{p+1}$.

12 pages