Contents Online

# Asian Journal of Mathematics

## Volume 8 (2004)

### Number 2

### Maps between B^{n} and B^{N} with geometric rank k_{0} ≤ n - 2 and minimum N

Pages: 233 – 258

DOI: http://dx.doi.org/10.4310/AJM.2004.v8.n2.a4

#### Authors

#### Abstract

Let B^{n} = {z ∆ Cn : |z| < 1} be the unit ball in C_{n}. The problem of classifying proper holomorphic mappings between B^{n} 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(B^{n},B^{N}) the collection of proper holomorphic mappings from B^{n} to B^{N}. It is known [A 1977] that any map F ∆ Prop(B^{n},B^{n}) must be biholomorphic and must be equivalent to the identity map. Here we say that f, g ∆Prop(B^{n},B^{N}) are equivalent if there are automorphisms σ ∆ Aut(B^{n}) and τ ∆ Aut(B^{N})) such that * f = τ ∘ g ∘ σ*. For general N > n, the discovery of inner functions indicates that Prop(B^{n},B^{N}) is too complicated to be classified. Hence we may focus on Rat(B^{n},B^{N}), the collection of all rational proper holomorphic mappings from B^{n} to B^{N}).