A generalization of half-plane mappings to the ball in ℂⁿ

Abstract

Let F F be a normalized ( F ( 0 ) = 0 F(0)=0 , D F ( 0 ) = I DF(0)=I ) biholomorphic mapping of the unit ball B ⊆ C n B \subseteq \mathbb C^n onto a convex domain Ω ⊆ C n \Omega \subseteq \mathbb C^n that is the union of lines parallel to some unit vector u ∈ C n u \in \mathbb C^n . We consider the situation in which there is one infinite singularity of F F on ∂ B \partial B . In one case with a simple change-of-variables, we classify all convex mappings of B B that are half-plane mappings in the first coordinate. In the more complicated case, when u u is not in the span of the infinite singularity, we derive a form of the mappings in dimension n = 2 n=2 .