Finitely Smooth Reinhardt Domains with Non-compact Automorphism Group

We give a complete description of bounded Reinhardt domains of finite boundary smoothness that have non-compact automorphism group. As part of this program, we show that the classification of domains with non-compact automorphism group and having only finite boundary smoothness is considerably more complicated than the classification of such domains that have infinitely smooth boundary. Let D ⊂ C be a bounded domain, and suppose that the group Aut(D) of holomorphic automorphisms of D is non-compact in the topology of uniform convergence on compact subsets of D. This means that there exist points q ∈ ∂D, p ∈ D and a sequence {fj} ⊂ Aut(D) such that fj(p) → q as j → ∞. We also assume that D is a Reinhardt domain, i.e. that the standard action of the n-dimensional torus T on C, zj 7→ e jzj , φj ∈ R, j = 1, . . . , n, leaves D invariant. In [FIK1] we gave a complete classification of bounded Reinhardt domains with non-compact automorphism group and C-smooth boundary. For the sake of completeness we quote the main result of [FIK1] below: Theorem 1. If D is a bounded Reinhardt domain in C with C-smooth boundary, and if Aut(D) is not compact then, up to dilations and permutations of coordinates, D is a domain of the form