Holomorphic mapping of products of annuli in Cⁿ

Eric Bedford

Vol. 87 (1980), No. 2, 271–281

DOI: 10.2140/pjm.1980.87.271

Abstract

Let Ω₁,Ω₂ ⊂ Cⁿ be bounded pseudoconvex Reinhardt domains with the property that z₁⋯z_n≠0 for all (z₁,⋯,z_n) ∈Ω_j. A holomorphic mapping f : Ω₁ → Ω₂ is discussed in terms of the induced mapping on homology f_∗ : H₁(Ω₁,R) → H₁(Ω₂,R). Specifically, there is a norm on H₁(Ω_j,R) which must decrease under f_∗. As a consequence we prove that a domain Ω as above is rigid in the sense of H. Cartan: if f : Ω → Ω is holomorphic and f_∗ : H₁(Ω,R) → H₁(Ω,R) is nonsingular, then f is an automorphism.

Mathematical Subject Classification 2000

Primary: 32A07

Secondary: 32H99

Milestones

Received: 24 April 1978

Published: 1 April 1980