PreprintCC BY 4.0arXiv
Length control for foliations by Jordan curves via smooth quasiconformal maps
Abstract
Let Ωsubsetneqℂ be a non-empty simply connected domain and 0<p<2. Bouchala and Koskela [Proc. Amer. Math. Soc. (2024)] proved that there exists a quasiconformal map f:𝔻→Ω such that ∫01ℓ(f(r S1))p dr<∞, and asked whether this conclusion holds at p=2, or even for every 0<p<2+ε with some ε>0. We answer this question affirmatively with ε=2/157 by constructing a single smooth $5$-quasiconformal map f:𝔻→Ω satisfying this integrability condition simultaneously for every 0<p<2+2/157.
Related
- Sobolev homeomorphisms and Jordan curves with positive area
- Liouville regular Jordan curves
- Rectangular Pegs on Jordan Curves of Finite p-Variation
- Isomorphism classes and geometric classification of three-dimensional Jordan algebras and the length function
- Geometric index theorems for holomorphic foliations and curves
BibTeX
@misc{pandey2026length,
title = {{Length control for foliations by Jordan curves via smooth quasiconformal maps}},
author = {Abhishek Pandey},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21009},
eprint = {2610.21009},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21009},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Length control for foliations by Jordan curves via smooth quasiconformal maps
AU - Abhishek Pandey
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.21009
UR - https://arxiv.org/abs/2610.21009
AB - Let Ωsubsetneqℂ be a non-empty simply connected domain and 0<p<2. Bouchala and Koskela [Proc. Amer. Math. Soc. (2024)] proved that there exists a quasiconformal map f:𝔻→Ω such that ∫_0^1ℓ(f(r S^1))^p dr<∞, and asked whether this conclusion holds at p=2, or even for every 0<p<2+ε with some ε>0. We answer this question affirmatively with ε=2/157 by constructing a single smooth $5$-quasiconformal map f:𝔻→Ω satisfying this integrability condition simultaneously for every 0<p<2+2/157.
ER -