On the inverse absolute continuity of quasiconformal mappings on hypersurfaces
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 Väisälä and Astala--Bonk--Heinonen.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2018
- DOI:
- 10.48550/arXiv.1810.05916
- arXiv:
- arXiv:1810.05916
- Bibcode:
- 2018arXiv181005916N
- Keywords:
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- Mathematics - Complex Variables;
- 30C65;
- 30L10
- E-Print:
- 21 pages, 2 figures, minor edits based on referee report. To appear in Amer. J. Math