Oeljeklaus, K.; Youssfi, E. H. Proper holomorphic mappings and related automorphism groups. (English) Zbl 0942.32019 J. Geom. Anal. 7, No. 4, 623-636 (1997). Let \(D\) be a simply connected strictly pseudoconvex bounded domain in \({\mathbb C}^n\) with smooth boundary and \(M\) an \(n\)-dimensional connected complex manifold. For any proper holomorphic map \(f:D\rightarrow M\), it was proved by E. Bedford and S. Bell [Math. Ann. 272, No. 4, 505-518 (1985; Zbl 0582.32024)] that there is a finite subgroup \(G\) of \(Aut_{\mathcal O}(D)\) such that \(f^{-1}\circ f(z)=\{g(z): g\in G\}\) for any \(z\in \Omega\). In this paper, the authors first show that the groups \[ \operatorname{Aut}_{\mathcal O}(M)={\mathcal N}/G,\;\;\operatorname{Aut}^0_{\mathcal O}(M)={\mathcal Z}^0/G\cap {\mathcal Z}^0 \] where \({\mathcal N}\) and \({\mathcal Z}\) denote the normalized and the centralizer of \(G\) in \(\operatorname{Aut}_{\mathcal O}(D)\). Indeed, they also show that there is a representation from \(\operatorname{Aut}_{\mathcal O}(M) \rightarrow \operatorname{Aut}_{\mathcal O}(D)\), and that if \(M\) is homogeneous, then \(f\) is biholomorphic and both \(M\) and \(D\) are biholomorphic to the ball. Some regularity and rigidity properties of proper holomorphic maps are also established. In particular, they solve a question raised by K. T. Hahn and P. Pflug [Monatsh. Math. 105, No. 2, 107-112 (1988; Zbl 0638.32005)] regarding the nonexistence of proper holomorphic maps between the Euclidean ball and the complex minimal ball of \({\mathbb C}^n\). Reviewer: Shanyu Ti (Houston) Cited in 6 Documents MSC: 32H35 Proper holomorphic mappings, finiteness theorems 32M05 Complex Lie groups, group actions on complex spaces Keywords:automorphism groups; simply connected strictly pseudoconvex domains; proper mappings Citations:Zbl 0582.32024; Zbl 0638.32005 PDFBibTeX XMLCite \textit{K. Oeljeklaus} and \textit{E. H. Youssfi}, J. Geom. 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