Quasiconformal geometry and removable sets for conformal mappings

TL;DR

This paper investigates when metric spaces defined by conformal weights that vanish on certain sets are quasiconformally equivalent to planar domains, linking this to the concept of removable sets for conformal mappings.

Contribution

It provides a characterization of such metric spaces in terms of removable sets and explores factorization of quasiconformal maps involving bi-Lipschitz maps.

Findings

Characterization of when these metric spaces are quasiconformally equivalent to planar domains.

Conditions under which quasiconformal maps can be factored through bi-Lipschitz maps.

Insights into the structure of removable sets for conformal mappings.

Abstract

We study metric spaces defined via a conformal weight, or more generally a measurable Finsler structure, on a domain $\Omega \subset \mathbb{R}^2$ that vanishes on a compact set $E \subset \Omega$ and satisfies mild assumptions. Our main question is to determine when such a space is quasiconformally equivalent to a planar domain. We give a characterization in terms of the notion of planar sets that are removable for conformal mappings. We also study the question of when a quasiconformal mapping can be factored as a 1-quasiconformal mapping precomposed with a bi-Lipschitz map.