On the Uniqueness of the Norton-Sullivan Quasiconformal extension

TL;DR

This paper proves the uniqueness of the Norton-Sullivan quasiconformal extension map for bi-Lipschitz functions, showing it is essentially the only such locally linear extension, up to a group action, and identifies other similar extensions.

Contribution

The paper establishes the uniqueness of the Norton-Sullivan extension map among locally linear extensions, and uncovers additional extensions within the same orbit.

Findings

The Norton-Sullivan extension is unique up to a group action.

Other extension maps similar to Norton-Sullivan exist within the same orbit.

The study characterizes the structure of all such extension maps.

Abstract

We show that the extension map \[ \mathcal{E}_{NS}(f)(z)=\frac{f(x+y)+f(x-y)}{2}+i\frac{f(x+y)-f(x-y)}{2}\mbox{ for all }z=x+iy\in\mathbb{H}\,, \] defined by Norton and Sullivan in '96, is the only locally linear extension map taking bi-Lipschitz functions on $\mathbb{R}$ to quasiconformal functions on $\mathbb{H}$, modulo the action of a group isomorphic to the linear group. In fact, we discovered many other extension like this one (lying in the orbit of such group action), such as: $f(x)\mapsto f(x)+i(f(x)-f(x-y))$.