Examples of domains with non-compact automorphism groups (Q5961478)

In this paper, the authors give an example of a bounded pseudoconvex domain \(D\) in \(\mathbb{C}^3\) with a real analytic boundary and with a noncompact automorphism group, but which is not holomorphically equivalent to a Reinhardt domain. Explicitly, \(D= \{|z_1 |^2 +|z_2 |^4+ |z_3 |^4+4 \text{Re} (z_2 \overline{z_3})^2 <1\}\). This domain was at first given by Bedford and Pincuck. The author describes \(\Aut(D)\) and proves that it is not compact. Their result is related to the fact that the isotropy group at the origin is of dimension two, contrary to the case of a bounded complete Reinhardt case where this group is of dimension at least three. This paper suggests that the situation of bounded domains of dimension three with a smooth (or even real-analytic boundary) and non-compact automorphism groups, is perhaps very wild. This is in contrast with the analogous problem in dimension two. In this case, results of Bedford and Pincuck lead to think that these domains are equivalent to some standard Reinhardt domains.NEWLINENEWLINENEWLINEIn their paper, the authors give also examples of bounded domains in \(\mathbb{C}^2\) with a smooth boundary outside a point but with a noncompact automorphism group.