Geometry of domains in \(\mathbb C^{n}\) with noncompact automorphism groups (Q1030956)

Within the last forty years many results have been achieved in complex analysis which give rather deep insight into the relationship between the geometry of manifolds and their groups of biholomorphic automorphisms. This is in particular true for domains in \(\mathbb C^n\). In the present paper, the authors cite and comment several theorems from this context, relating some of them to their own work, published in different articles [Vietnam J. Math. 37, No.~1, 67--79 (2009; Zbl 1181.32031)]. The most recent results by other authors which they discuss are the following ones: Every connected real Lie group can be realized as the full automorphism group of some strictly pseudoconvex bounded domain [\textit{J. Winkelmann}, Comment. Math. Helv. 79, No.~2, 285--299 (2004; Zbl 1056.32022)]; for compact groups see [\textit{E. Bedford} and \textit{J. Dadok}, Comment. Math. Helv. 62, 561--572 (1987; Zbl 0647.32027); and \textit{R. Saerens} and \textit{W. Zame}, Trans. Am. Math. Soc. 301, 413--429 (1987; Zbl 0621.32025)]. The complex polydisc \(\triangle^n\) can be characterized by its automorphism group [\textit{A. V. Isaev} J. Geom. Anal. 18, No.~3, 795--799 (2008; Zbl 1146.32009)] for a weaker version see, \textit{A. Kodama} and \textit{S. Shimizu}, Mich. Math. J. 56, No.~1, 173--181 (2008; Zbl 1171.32011)]. The result of Isaev is not correctly cited in the present article (Theorem \(4.7\)). The main assumption is missing, namely that \(\Aut(M)\) and \(\Aut(\triangle^n)\) should be isomorphic as topological groups equipped with the compact-open topology.