Calculus of Variations and Geometric Measure Theory
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D. Ntalampekos - M. Romney

On the inverse absolute continuity of quasiconformal mappings on hypersurfaces

created by romney on 21 Dec 2018

[BibTeX]

Submitted Paper

Inserted: 21 dec 2018

Year: 2018

ArXiv: 1810.05916 PDF

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.

Tags: GeoMeG
Keywords: quasiconformal mapping, absolute continuity

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