Metric Rectifiability of ℍ-regular Surfaces with Hölder Continuous Horizontal Normal
| Autor: | Katrin Fässler, Daniela Di Donato, Tuomas Orponen |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | International Mathematics Research Notices. 2022:17909-17975 |
| ISSN: | 1687-0247 1073-7928 |
| DOI: | 10.1093/imrn/rnab227 |
| Popis: | Two definitions for the rectifiability of hypersurfaces in Heisenberg groups $\mathbb{H}^n$ have been proposed: one based on ${\mathbb{H}}$-regular surfaces and the other on Lipschitz images of subsets of codimension-$1$ vertical subgroups. The equivalence between these notions remains an open problem. Recent partial results are due to Cole–Pauls, Bigolin–Vittone, and Antonelli–Le Donne. This paper makes progress in one direction: the metric Lipschitz rectifiability of ${\mathbb{H}}$-regular surfaces. We prove that ${\mathbb{H}}$-regular surfaces in $\mathbb{H}^{n}$ with $\alpha $-Hölder continuous horizontal normal, $\alpha> 0$, are metric bilipschitz rectifiable. This improves on the work by Antonelli–Le Donne, where the same conclusion was obtained for $C^{\infty }$-surfaces. In $\mathbb{H}^{1}$, we prove a slightly stronger result: every codimension-$1$ intrinsic Lipschitz graph with an $\epsilon $ of extra regularity in the vertical direction is metric bilipschitz rectifiable. All the proofs in the paper are based on a new general criterion for finding bilipschitz maps between “big pieces” of metric spaces. |
| Databáze: | OpenAIRE |
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