Complex \(n\)-dimensional manifolds with a real \(n^2\)-dimensional automorphism group
Suppose \(M\) is a Kobayashi hyperbolic complex manifold of (complex) dimension \(n\geq 2\). Then \(\Aut (M)\) is a Lie group of (real) dimension less than or equal to \(n^2+2n\). It is classical that \(\dim \Aut(M) = n^2+2n\) implies \(M\) is biholomorphic to \({\mathbb B}_{n}\). If \(\dim \Aut(M) > n^2\), then \textit{W. Kaup} [Invent. Math. 3, 43--70 (1967; Zbl 0157.13401)] showed that \(M\) is homogeneous. These results were refined by \textit{A. Isaev} and \textit{S. Krantz} in a series of papers [Adv. Math. 146, 1--38 (1999; Zbl 1040.32019), J. Korean Math. Soc. 37, 297--308 (2000; Zbl 0965.32005), J. Reine Angew. Math. 534, 187--194 (2001; Zbl 0974.32018)]. If \(\dim \Aut(M) \geq n^2+3\), then \(M\) is biholomorphic to \({\mathbb B}_{n}\); if \(\dim \Aut(M) = n^2+2\), then \(M\) is biholomorphic to \({\mathbb B}_{n-1}\times{\mathbb B}_{1}\); and if \(\dim \Aut(M) = n^2+1\), then \(n=3\) and \(M\) is biholomorphic to a certain Siegel half space. In the present paper the authors consider a simply connected, complete Kobayashi hyperbolic manifold \(M\) of dimension \(n\geq 2\) with \(\dim \Aut(M) = n^2\). There are two cases, \(M\) is homogeneous or not. If \(M\) is homogeneous, either \(n=3\) and \(M\) is biholomorphic to the tridisk \({\mathbb D}^3\) or \(n=4\) and \(M\) is biholomorphic to \( {\mathbb B}_{2} \times {\mathbb B}_{2}\). When \(M\) is not homogeneous, the authors show that \(M\) is then biholomorphic to a Thullen domain \(\{ (z_{1},\ldots,z_{n})\in{\mathbb C}^n | | z_{1}| ^2 + \ldots + | z_{n-1}| ^2 + | z_{n}| ^r < 1 \}\), where \(r\) is some positive real number not equal to \(2\). One should also note that the non-homogeneous case has been studied assuming that the manifold is a hyperbolic Reinhardt domain, see the paper of \textit{J. A. Gifford, A. V. Isaev} and \textit{S. G. Krantz} [Illinois J. Math. 44, 602--618 (2000; Zbl 0968.32001)]. The main tool used in the proof is an analysis of the tangent representation of the isotropy subgroup action.
scientific article; zbMATH DE number 2163717
scientific article