A generalization of Cartan's theorem to proper holomorphic mappings (Q1099303)

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.