On the inverse absolute continuity of quasiconformal mappings on hypersurfaces

TL;DR

This paper constructs specific quasiconformal mappings in three dimensions that demonstrate the failure of inverse absolute continuity on hypersurfaces, solving a longstanding open problem and answering related questions in the field.

Contribution

It provides the first example of quasiconformal mappings with non-absolutely continuous inverse on hypersurfaces, addressing a key open problem in geometric function theory.

Findings

Constructed quasiconformal mappings with positive measure sets mapping to measure zero sets

Resolved the open problem of inverse absolute continuity on hypersurfaces

Answered questions posed by V"ais"al"a and Astala--Bonk--Heinonen

Abstract

We construct quasiconformal mappings $f\colon \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ for which there is a Borel set $E \subset \mathbb{R}^2 \times \{0\}$ of positive Lebesgue $2$-measure whose image $f(E)$ has Hausdorff $2$-measure zero. This gives a solution to the open problem of inverse absolute continuity of quasiconformal mappings on hypersurfaces, attributed to Gehring. By implication, our result also answers questions of V\"ais\"al\"a and Astala--Bonk--Heinonen.