Non-removable sets for quasiconformal and locally biLipschitz mappings in R^3

TL;DR

This paper constructs a totally disconnected set in R^3 with Hausdorff dimension 2 that is non-removable for quasiconformal and locally biLipschitz homeomorphisms, demonstrating limits of removability in geometric function theory.

Contribution

It provides explicit examples of non-removable sets in R^3 for both quasiconformal and locally biLipschitz mappings, expanding understanding of geometric removability.

Findings

Existence of a non-removable set with Hausdorff dimension 2 in R^3.

Such sets are totally disconnected.

The construction applies to both quasiconformal and locally biLipschitz homeomorphisms.

Abstract

We give an example of a totally disconnected set E in R^3 which is not removable for quasiconformal homeomorphisms, i.e., there is a homeomorphism f of R^3 to itself which is quasiconformal off E, but not quasiconformal on all of R^3. The set E may be taken with Hausdorff dimension 2. The construction also gives a non-removable set for locally biLipschitz homeomorphisms.