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arXiv · 1810.05916

On the inverse absolute continuity of quasiconformal mappings on hypersurfaces

Dimitrios Ntalampekos·Matthew Romney
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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.

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Dimitrios Ntalampekos, Matthew Romney. 2018-10-13. On the inverse absolute continuity of quasiconformal mappings on hypersurfaces. https://doi.org/10.1353/ajm.2021.0041

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