On the Uniqueness of the Norton-Sullivan Quasiconformal extension
José Afonso Barrionuevo, Felipe Gonçalves, José Victor Medeiros, Lucas Oliveira
TL;DR
This work investigates the uniqueness of the Norton–Sullivan quasiconformal extension by classifying all admissible extensions that are locally linear near the boundary. The authors introduce a two-parameter family $\mathcal{E}_{a,\alpha}$ of extensions and show that, after applying a natural group action, any extension in the admissible class that is linear at the identity (or has a vanishing second variation) must be one of the $\mathcal{E}_{a,\alpha}$, with $\mathcal{E}_{1,2}=\mathcal{E}_{NS}$. The key strategy expresses extensions near $i$ in terms of Radon measures $\mu,\nu$, derives a functional identity via perturbations in $BL_+({\mathbb R})$, and uses a variational analysis to force $\mu,\nu$ to be supported on at most two points, thereby identifying the extension uniquely up to the group action. The results illuminate the rigidity of boundary-to-domain extension maps within this structured setting and pose open directions, including alternative formulations of the defining conditions and a Cauchy problem viewpoint for Norton–Sullivan–like extensions.
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: $f(x)\mapsto f(x)+i(f(x)-f(x-y))$.