On the Dimensions of the Automorphism Groups of Hyperbolic Reinhardt Domains

Let D be a domain (a connected open set) in C, n ≥ 2. Denote by Aut(D) the group of holomorphic automorphisms of D; that is, Aut(D) is the group under composition of all biholomorphic self-maps of D. If D is bounded or, more generally, Kobayashi-hyperbolic, then the group Aut(D) with the topology of uniform convergence on compact subsets of D is in fact a finitedimensional Lie group (see [Ko]). We note that this Lie group is always a real group but never a complex Lie group (except for the case of zerodimensional groups). Thus, when we specify the dimension of this group, we shall always be speaking of its real dimension. By contrast, when we speak of the dimension of the domain on which it acts, we shall be referring to complex dimension. We are interested in characterizing a domain by its automorphism group. Much work has been done on classifying domains with non-compact automorphism group (see [IK2] for a detailed exposition). In this paper we concentrate on the dimension of Aut(D). Namely, we are interested in the following question: to what extent does the dimension of the automorphism group determine the domain?