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On the inverse absolute continuity of quasiconformal mappings on hypersurfaces

Matthew RomneyDimitrios Ntalampekos

subject

Pure mathematicsMathematics::Complex VariablesMathematics - Complex VariablesGeneral MathematicsImage (category theory)Open problem010102 general mathematicsHausdorff spaceZero (complex analysis)InverseAbsolute continuityLebesgue integration01 natural sciences30C65 30L10funktioteoriasymbols.namesakeFOS: MathematicssymbolsMathematics::Metric GeometryComplex Variables (math.CV)0101 mathematicsBorel setMathematics

description

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.

yearjournalcountryeditionlanguage
2018-10-13
10.1353/ajm.2021.0041http://arxiv.org/abs/1810.05916