Maps between Bn and BN with geometric rank k0 ≤ n - 2 and minimum N

Abstract

Let Bⁿ = {z ∆ Cn : |z| < 1} be the unit ball in C_n. The problem of classifying proper holomorphic mappings between Bⁿ and B^N has attracted considerable attention (see [Fo 1992] [DA 1988] [DA 1993] [W 1979] [H 1999][HJ 2001] for extensive references) since the work of Poincare [P 1907][T 1962] and Alexander [A 1977]. Let us denote by Prop(Bⁿ,B^N) the collection of proper holomorphic mappings from Bⁿ to B^N. It is known [A 1977] that any map F ∆ Prop(Bⁿ,Bⁿ) must be biholomorphic and must be equivalent to the identity map. Here we say that f, g ∆Prop(Bⁿ,B^N) are equivalent if there are automorphisms σ ∆ Aut(Bⁿ) and τ ∆ Aut(B^N)) such that f = τ ∘ g ∘ σ. For general N > n, the discovery of inner functions indicates that Prop(Bⁿ,B^N) is too complicated to be classified. Hence we may focus on Rat(Bⁿ,B^N), the collection of all rational proper holomorphic mappings from Bⁿ to B^N).