The constant curvature property of the Wu invariant metric. (Q1880069)

The object of the article under review is the hermitian metric introduced by \textit{H. H.Wu} [Several complex variables, Proc. Mittag-Leffler Inst., Stockholm/Swed. 1987-88, Math. Notes 38, 640--682 (1993; Zbl 0773.32017)]. In their main results the authors prove curvature properties of this metric, namely: Theorem 1: Let \(\Omega\) be a domain obtained by intersecting the ball \(B^n\) with an open set. Denote the boundaries of \(\Omega\) and \(B\) by \(\partial \Omega\) and \(\partial B\), respectively. There exists a neighborhood \(V\) of \(\partial \Omega \cap \partial B^n\), such that the Wu metric of \(\Omega\) is Kähler with constant negative holomorphic curvature in \(V\cap \Omega\). Theorem 2: Let \(\widetilde E = \{ (z_1,\dots,z_n) \in \mathbb C^n\mid | z_1| ^{2m_1}+\dots + | z_n| ^{2m_n}<1\}\), where \(m_1,\dots,m_n\) are integers \(\geq 2\). If \(p\) is a strongly pseudoconvex boundary point of \(\widetilde E\), then there exists a neighorhood \(V_p\) of \(p\), such that the Wu metric of \(\widetilde E\) is Kähler with constant negative holomorphic curvature in \(V_p \cap \widetilde E\).