Curvature and proper holomorphic mappings between bounded domains in \({\mathbb{C}}^ n\)
This is an expanded version of the authors' paper ``A note on pseudoconvexity and proper holomorphic mappings'' [Proc. Am. Math. Soc. 102, No.3, 646-650 (1988; Zbl 0657.53037)]. Let \(D_ 1\) and \(D_ 2\) be bounded domains in \({\mathbb{C}}^ n\). Let \(P(D_ 1,D_ 2)\) (resp. \(P_ 0(D_ 1,D_ 2))\) denote the set of proper holomorphic maps (resp. unbranched proper holomorphic maps) from \(D_ 1\) to \(D_ 2\). The main result is the following: suppose there exist a strongly pseudoconvex boundary point \(p\in \partial D_ 2\), a point \(x\in D_ 1\), and a sequence \(\{f_ j\}\subset P_ 0(D_ 1,D_ 2)\) such that \(f_ j(x)\to p\). Then \(D_ 1\) and \(D_ 2\) are biholomorphic to the unit ball \(B_ n\) in \({\mathbb{C}}^ n.\) From this the authors deduce that for bounded strongly pseudoconvex domains in \({\mathbb{C}}^ n\), \(n\geq 2\), \(P(D_ 1,D_ 2)\) is noncompact iff both \(D_ 1\) and \(D_ 2\) are biholomorphic to \(B_ n\). There is one step in the proof which the authors do not discuss here but which is nevertheless valid [the authors, An appendix to ``Curvature and proper holomorphic mappings between bounded domains in \({\mathbb{C}}^ n\)'', to appear]: a sequence \(\{f_ j\}\subset P(D_ 1,D_ 2)\) can never converge uniformly on compact subsets to a non-proper holomorphic mapping f: \(D_ 1\to D_ 2\).
scientific article; zbMATH DE number 4165304
scientific article