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

TL;DR

This paper investigates shearing holomorphic maps of the unit ball, constructing a specific univalent map onto a Runge domain in complex n-space that cannot be embedded into a Loewner chain with range c^n.

Contribution

It introduces a novel example of a univalent map from the unit ball to a Runge domain that defies embedding into a Loewner chain with full complex space range.

Findings

Constructed a univalent map onto a Runge domain in c^n

Demonstrated the map cannot be embedded into a Loewner chain with c^n range

Analyzed properties of shearing holomorphic maps of the unit ball

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 $\mathbb C^n$ which however cannot be embedded into a Loewner chain whose range is $\mathbb C^n$.