An application of Einstein Kähler metrics to proper holomorphic mappings between pseudoconvex domains (Q1127813)

This paper is devoted to the following conjectural generalization of a result by the author [Invent. Math. 41, 253-257 (1977; Zbl 0385.32016)]. Let \(D_1\) and \(D_2\) be two bounded strongly pseudoconvex domains with smooth boundary in \(\mathbb{C}^n\), \(n \geq 2\). Then both \(D_1\) and \(D_2\) are biholomorphic to an euclidean ball if and only if the set \(P(D_1, D_2)\) of all proper holomorphic mappings from \(D_1\) to \(D_2\) is a non-compact subset in the set \(C(D_1, D_2)\) of all continuous mappings from \(D_1\) to \(D_2\) relative to compact open topology. The main result of the paper under review is that if \(P(D_1,D_2) \subset C(D_1, D_2)\) is non-compact, then both \(D_1\) and \(D_2\) are covered holomorphically by the euclidean ball. The author notes that the proof essentially uses the properties of the invariant Einstein Kähler metric constructed in \textit{S. Y. Cheng} and \textit{S. T. Yau} [Commun. Pure Appl. Math. 33, 507-544 (1980; Zbl 0506.53031)].