Diffeomorphisms between Siegel domains of the first kind preserving the holomorphic automorphism groups and applications (Q363261)

Let \(D\), \(D'\) be Siegel domains of the first kind in \(\mathbb C^n\) and denote by \(G = \text{Aut}^o(D)\) and \(G' = \text{Aut}^o(D')\) the identity components of their groups of automorphisms. The authors give criterions for existences of real analytic diffeomorphisms between \(D\) and \(D'\), based on algebraic properties of \(G\) and \(G'\). In particular, they prove that if one of the two domains is homogeneous and \(G\) and \(G'\) are isomorphic as topological groups, then both domains are homogeneous and there exists a real analytic diffeomorphism \(F: D \longrightarrow D'\) of a very special form. They also consider additional conditions that imply the existence of biholomorphisms. They obtain that if \(n \leq 3\), then \(D\) and \(D'\) are biholomorphic if and only if \(G\) and \(G'\) are isomorphic as topological groups.