Estimate of the Einstein-Kähler metric on a weakly pseudoconvex domain in \(\mathbb{C}^ 2\) (Q5961421)

\textit{S. Y. Cheng} and \textit{S. T. Yau} [Commun. Pure Appl. Math. 33, 507-544 (1980; Zbl 0506.53031)] proved that there is an Einstein-Kähler metric on a weakly pseudoconvex domain. Furthermore, \textit{N. Mok} and \textit{S. T. Yau} [Proc. Symp. Pure Math. 39, Part 1, Bloomington/Indiana 1980, 41-59 (1983; Zbl 0526.53056)] proved that there exists an Einstein-Kähler metric on a \(C^2\) domain if and only if this domain is pseudoconvex. For a strictly pseudoconvex domain, Cheng and Yau also proved that the Einstein-Kähler metric is equivalent to the metric \(\bigl(-\frac {\partial^2\ln\phi} {\partial z_i\partial\overline{z}_j} (z))\), where \(\phi\) is the defining function of the domain. The only result on the boundary behavior of the Einstein-Kähler metric for a non-strictly pseudoconvex domain is due to \textit{J. S. Bland} [Mich. Math. J. 33, 209-220 (1986; Zbl 0602.32005)]. He describes the boundary behavior of the Einstein-Kähler metric for the domain \(\Omega_p =\{(z,w)\in \mathbb{C}^{n+1}\mid |z|^2 + |w|^{2p}<1\}\), where \(p\) is a positive integer. The purpose of this paper is to give an estimate of the Einstein-Kähler metric of a weakly pseudoconvex domain of finite type in \(\mathbb{C}^2\). We use the Monge-Ampère equation associated to the Bergman metric, instead of the metric \((g_{i\overline{j}}=\bigl(-\frac{\partial^2 \ln\phi}{\partial z_i\partial\overline{z}_j} (z)\bigr)\) defined by the defining function, to get the Einstein-Kähler metric and then the desired estimate.