A generalization of Cartan's theorem to proper holomorphic mappings
The theorem of \textit{H. Cartan} [Math. Z. 35, 760-773 (1932; Zbl 0004.40602)] says: Consider a sequence of biholomorphic mappings \(f_ i:\Omega_ 1\to \Omega_ 2\) between bounded domains in \({\mathbb{C}}^ n,\) which converges uniformly on compact subsets of \(\Omega_ 1\) to a mapping f. Then: (A) f is a biholomorphic mapping of \(\Omega_ 1\) onto \(\Omega_ 2\), iff (B) \(f(\Omega_ 1)\) is not a subset of \(b\Omega_ 2\), iff (C) the jacobian determinant det [f'(z)] is not identically zero on \(\Omega_ 1.\) The author gives an example in \({\mathbb{C}}^ 1 \)to show that one can not replace ``biholomorphic'' by ``proper'' in this theorem, and proves a theorem in which under some additional hypotheses this is possible. The author's theorem says: Let \(\Omega_ 1\) and \(\Omega_ 2\) be additionally pseudoconvex and have bounded plurisubharmonic exhaustion functions, and let \(f_ i\) be proper holomorphic and additionally have multiplicities bounded by some m. Then : (A') f is a proper mapping of \(\Omega_ 1\) onto \(\Omega_ 2\), iff Cartan's (B) holds, iff Cartan's (C) holds. Moreover, if (A') takes place, then the multiplicity of f is not larger than m. All the necessary notions are defined in the text. The author describes how to use the theorem to examine which domains in \({\mathbb{C}}^ n \)have no proper holomorphic self-mappings which are not biholomorphic. He also asks (a) if his theorem holds for bounded taut domains, and (b) if the exhaustion functions are really needed in his theorem.
sequence of proper holomorphic mappings converging uniformly on compact subsets
scientific article; zbMATH DE number 4040302