On the inverse absolute continuity of quasiconformal mappings on hypersurfaces

Abstract

We construct quasiconformal mappings $f:{\mathbb{R}}^3\to {\mathbb{R}}^3$ 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 Väisälä and Astala--Bonk--Heinonen.

21 pages, 2 figures, minor edits based on referee report. To appear in Amer. J. Math

References in corpus (1)

• Non-removability of Sierpinski carpets

Cited by in corpus (1)

• Polyhedral approximation of metric surfaces and applications to uniformization