Shearing maps and a Runge map of the unit ball which does not embed into a Loewner chain with range $\C^n$

Abstract

In this paper we study the class of ``shearing'' holomorphic maps of the unit ball of the form $(z,w)\mapsto (z+g(w), w)$. Besides general properties, we use such maps to construct an example of a normalized univalent map of the ball onto a Runge domain in $\C^n$ which however cannot be embedded into a Loewner chain whose range is $\C^n$.