## Asian Journal of Mathematics

### 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 Cn. The problem of classifying proper holomorphic mappings between Bn and BN 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,BN) the collection of proper holomorphic mappings from Bn to BN. 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,BN) are equivalent if there are automorphisms σ ∆ Aut(Bn) and τ ∆ Aut(BN)) such that f = τ ∘ g ∘ σ. For general N > n, the discovery of inner functions indicates that Prop(Bn,BN) is too complicated to be classified. Hence we may focus on Rat(Bn,BN), the collection of all rational proper holomorphic mappings from Bn to BN).

#### Article information

Source
Asian J. Math., Volume 8, Number 2 (2004), 233-258.

Dates
First available in Project Euclid: 24 June 2004

Permanent link to this document
https://projecteuclid.org/euclid.ajm/1088090061

Mathematical Reviews number (MathSciNet)
MR2129536

#### Citation

Yi, Shanyu; Xu, Dekang. Maps between B n and B N with geometric rank k 0 ≤ n - 2 and minimum N. Asian J. Math. 8 (2004), no. 2, 233--258. https://projecteuclid.org/euclid.ajm/1088090061