Metric Rectifiability of H-regular Surfaces with Hölder Continuous Horizontal Normal
Di Donato, D., Fässler, K., & Orponen, T. (2022). Metric Rectifiability of H-regular Surfaces with Hölder Continuous Horizontal Normal. International Mathematics Research Notices, 2022(22), 17909-17975. https://doi.org/10.1093/imrn/rnab227
Published inInternational Mathematics Research Notices
© The Author(s) 2021
Two definitions for the rectifiability of hypersurfaces in Heisenberg groups Hn have been proposed: one based on 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 H-regular surfaces. We prove that H-regular surfaces in Hn with α-Hölder continuous horizontal normal, α>0, are metric bilipschitz rectifiable. This improves on the work by Antonelli–Le Donne, where the same conclusion was obtained for C∞-surfaces. In H1, we prove a slightly stronger result: every codimension-1 intrinsic Lipschitz graph with an ϵ 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. ...
PublisherOxford University Press
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Related funder(s)Academy of Finland; European Commission
Funding program(s)Academy Research Fellow, AoF; Academy Project, AoF
The content of the publication reflects only the author’s view. The funder is not responsible for any use that may be made of the information it contains.
Additional information about fundingD.D.D. is partially supported by the Academy of Finland (Enrico Le Donne's grants 288501 `Geometry of sub-Riemannian groups' and 322898 `Sub-Riemannian geometry via metric-geometry and Lie-group theory') and by the European Research Council (Enrico Le Donne's ERC starting grant 713998 GeoMeG `Geometry of Metric Groups'); K.F and T.O are supported by the Academy of Finland (grants 321696 `Singular integrals, harmonic functions, and boundary regularity in Heisenberg groups' to K.F. and 309365 and 314172 `Quantitative rectifiability in Euclidean and non-Euclidean spaces' to T.O.). ...
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