Asian Journal of Mathematics

Volume 8 (2004)

Number 2

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

Pages: 233 – 258

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

Authors

Dekang Xu

Shanyu Yi

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).

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