Reinhardt Domains with Non-compact Automorphism Groups

We give an explicit description of smoothly bounded Reinhardt domains with noncompact automorphism groups. In particular, this description confirms a special case of a conjecture of Greene/Krantz. 0. Introduction Let D be a bounded domain in C, n ≥ 2. Denote by Aut(D) the group of holomorphic automorphisms of D. The group Aut(D) with the topology of uniform convergence on compact subsets of D is in fact a Lie group (see [Ko]). This paper is motivated by known results characterizing a domain by its automorphism group (see e.g. [R], [W], [BP]). More precisely, we assume that Aut(D) is not compact, i.e. there exist p ∈ D, q ∈ ∂D and a sequence {Fi} in Aut(D) such that Fi(p) → q as i → ∞. A point q ∈ ∂D with the above property is called a boundary accumulation point for Aut(D). An important issue for describing a domain D in terms of Aut(D) is the geometry of ∂D near a boundary accumulation point q (see e.g. [BP], [GK2]). In particular, we will be interested in the type of ∂D at q in the sense of D’Angelo [D’A], which measures the order of contact that complex varieties passing through q may have with ∂D. We note in passing that it is known that ∂D must be pseudoconvex at a boundary accumulation point [GK1]. It is desirable to have additional geometric information about boundary accumulation points. We will be discussing the following conjecture that can be found in [GK2]: if D is a bounded domain in C with C∞-smooth boundary and if q is a boundary accumulation point for Aut(D), then q is a point of finite type. For convex domains the conjecture was studied in [Ki]. Here we assume that D is a Reinhardt domain, i.e. a domain which the standard action of the n-dimensional torus T on C, zj 7→ ejzj , φj ∈ R, j = 1, . . . , n, (0.1) leaves invariant. The automorphism groups of bounded (and even hyperbolic) Reinhardt domains have been determined in [Su], [Sh2], [Kr]. We will use this description to prove the following classification result. Mathematics Subject Classification: 32A07, 32H05, 32M05