The Einstein-Kähler metric on \(\{| z| ^ 2+| w| ^{2p}<1\}\)
The aim of the paper is to describe the Einstein-Kähler metric for the domain \(D = \{(z,w)\in {\mathbb{C}}^ n\times {\mathbb{C}}: \| z\|^ 2 + | w|^{2p}< 1\}\) where p is a positive number. A formula for the metric is given in terms of a function determined by a differential equation. Moreover, the components of the curvature tensor for the metric are calculated and the behaviour of the metric near boundary points of special interest is analyzed. (The existence of Einstein-Kähler metrics in bounded pseudoconvex domains has been proved by \textit{N. Mok} and \textit{S. T. Yau} in Proc. Symp. Pure Math. 39, Part 1, 41-59 (1983; Zbl 0526.53056). The case of bounded pseudoconvex domains with \({\mathcal C}^ 2\)-boundary has been investigated by \textit{S. Y. Cheng} and \textit{S. T. Yau} in Commun. Pure Appl. Math. 33, 507-544 (1980; Zbl 0506.53031).)
Einstein-Kähler metrics in bounded pseudoconvex domains
scientific article; zbMATH DE number 3971183