On the automorphism groups of hyperbolic manifolds (Q2717082)

Let \(M\) be a connected hyperbolic complex manifold of dimension at least two. By classical results of Kobayashi, the automorphism group \(G\) of \(M\) is a real Lie group of dimension at most \(n^2+2n\). The authors continue their classification of \(M\) in terms of \(\dim(G)\). In earlier work [J. Korean Math. Soc. 37, No. 2, 297-308 (2000; Zbl 0965.32005)], they proved: NEWLINENEWLINENEWLINE(i) if \(\dim(G) \geq n^2 + 3\), then \(M\) is biholomorphically equivalent to the unit ball \(B^n \subset {\mathbb C}^n\), NEWLINENEWLINENEWLINE(ii) if \(\dim(G) = n^2 + 2\), then \(M\) is biholomorphically equivalent to \(B^{n-1} \times B^1\). The main result of this article says:NEWLINENEWLINENEWLINE(iii) if \(\dim(G) = n^2 + 1\), then \(M\) is biholomorphically equivalent to the \(3\)-dimensional Siegel space. NEWLINENEWLINENEWLINEMoreover, the authors give a new proof of (i) and obtain more generally: If \(M\) is any connected complex manifold of dimension \(n \geq 2\), and \(G \subset \text{ Aut}(M)\) is a real Lie group of dimension at least \(n^2 + 3\) acting properly on \(X\), then \(X\) is biholomorphically equivalent to either \(B^n\), \({\mathbb C}^n\) or the projective space \({\mathbb C\mathbb P}^n\).