ON PROPER HOLOMORPHIC MAPPINGS BETWEEN TWO EQUIDIMENSIONAL FBH-TYPE DOMAINS

Abstract

We introduce a new class of domains Dn,m(μ, p), called FBH-type domains, in ℂn × ℂm, where 0 < μ ∈ ℝ and p ∈ ℕ. In the special case of p = 1, these domains are just the Fock-Bargmann-Hartogs domains Dn,m(μ) in ℂn × ℂm introduced by Yamamori. In this paper we obtain a complete description of an arbitrarily given proper holomorphic mapping between two equidimensional FBH-type domains. In particular, we prove that the holomorphic automorphism group Aut(Dn,m(μ, p)) of any FBH-type domain Dn,m(μ, p) with p ≠ 1 is a Lie group isomorphic to the compact connected Lie group U(n) × U(m). This tells us that the structure of Aut(Dn,m(μ, p)) with p ≠ 1 is essentially different from that of Aut(Dn,m(μ)).