On the inverse absolute continuity of quasiconformal mappings on hypersurfaces

Dimitrios Ntalampekos; Matthew Romney

doi:10.1353/ajm.2021.0041

On the inverse absolute continuity of quasiconformal mappings on hypersurfaces

Dimitrios Ntalampekos
, Matthew Romney

Department of Mathematical Sciences

Stony Brook University

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

We construct quasiconformal mappings f: ℝ³ → ℝ³ for which there is a Borel set E ⊂ ℝ² ×{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.

Original language	English
Pages (from-to)	1633-1659
Number of pages	27
Journal	American Journal of Mathematics
Volume	143
Issue number	5
DOIs	https://doi.org/10.1353/ajm.2021.0041
State	Published - Oct 2021

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10.1353/ajm.2021.0041

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abstract = "We construct quasiconformal mappings f: ℝ3 → ℝ3 for which there is a Borel set E ⊂ ℝ2 ×\{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{\"a}is{\"a}l{\"a} and Astala-Bonk-Heinonen.",

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AB - We construct quasiconformal mappings f: ℝ3 → ℝ3 for which there is a Borel set E ⊂ ℝ2 ×{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.

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On the inverse absolute continuity of quasiconformal mappings on hypersurfaces

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