Proper holomorphic mappings in the special class of Reinhardt domains

Abstract

Abstract. A complete characterization of proper holomorphic mappings between domains from the class of all pseudoconvex Reinhardt domains in C 2 with the logarithmic image equal to a strip or a half-plane is given. 1. Statement of results We adopt here the standard notations from complex analysis. Given γ = (γ1, γ2) ∈ R 2 and z = (z1, z2) ∈ C 2 for which it makes sense we put |z γ | = |z1 | γ1 |z2 | γ2. The unit disc in C is denoted by D and the set of proper holomorphic mappings between domains D, G ⊂ C n is denoted by Prop(D, G). In this paper we deal with the pseudoconvex Reinhardt domains in C 2 whose logarithmic image is equal to a strip or a half-plane. Observe that such domains are always algebraically equivalent to domains of the form D α,r −,r +: = {z ∈ C 2: r − < |z α | < r +}, where α = (α1, α2) ∈ (R2)∗, 0 < r + < ∞, − ∞ < r − < r +. We say that Dα,r −,r+ is of the irrational type if α1/α2 ∈ R \\ Q. In the other case Dα,r −,r+ is said to be of the rational type. Recall that if r − < 0 < r +, α ∈ (R2)∗, then the domains Dα,r −,r+ are so-called elementary Reinhardt domains. Below we shall give a complete description of all proper holomorphic mappings between the domains D − α,r1,r+ 1 r + i < ∞, − ∞ < r − i < r+ i and D − β,r2,r+ 2 for arbitrary α, β ∈ (R 2) ∗ and 0 <, i = 1, 2. Similar problems were studied in some papers. In [Shi1] and [Shi2] the problem of holomorphic equivalence of elementary Reinhardt domains was considered. These results were partially extended by A. Edigarian and W. Zwonek. In the paper [Edi-Zwo] the authors gave a characterization of proper holomorphic mappings between elementary Reinhardt domains of the rational type.