A1 Journal article (refereed)
Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces (2020)

Antonelli, G., & Le Donne, E. (2020). Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces. Nonlinear Analysis : Theory, Methods and Applications, 200, Article 111983. https://doi.org/10.1016/j.na.2020.111983

JYU authors or editors

Publication details

All authors or editors: Antonelli, Gioacchino; Le Donne, Enrico

Journal or series: Nonlinear Analysis : Theory, Methods and Applications

ISSN: 0362-546X

eISSN: 1873-5215

Publication year: 2020

Volume: 200

Article number: 111983

Publisher: Elsevier

Publication country: United Kingdom

Publication language: English

DOI: https://doi.org/10.1016/j.na.2020.111983

Publication open access: Openly available

Publication channel open access: Partially open access channel

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/76297

Web address of parallel published publication (pre-print): https://arxiv.org/abs/1910.12812

Abstract

This paper is related to the problem of finding a good notion of rectifiability in sub-Riemannian geometry. In particular, we study which kind of results can be expected for smooth hypersurfaces in Carnot groups. Our main contribution will be a consequence of the following result: there exists a -hypersurface without characteristic points that has uncountably many pairwise non-isomorphic tangent groups on every positive-measure subset. The example is found in a Carnot group of topological dimension 8, it has Hausdorff dimension 12 and so we use on it the Hausdorff measure . As a consequence, we show that any Lipschitz map defined on a subset of a Carnot group of Hausdorff dimension 12, with values in , has negligible image with respect to the Hausdorff measure . In particular, we deduce that cannot be Lipschitz parametrizable by countably many maps each defined on some subset of some Carnot group of Hausdorff dimension 12. As main consequence we have that a notion of rectifiability proposed by S. Pauls is not equivalent to one proposed by B. Franchi, R. Serapioni and F. Serra Cassano, at least for arbitrary Carnot groups. In addition, we show that, given a subset of a homogeneous subgroup of Hausdorff dimension 12 of a Carnot group, every bi-Lipschitz map satisfies . Finally, we prove that such an example does not exist in Heisenberg groups: we prove that all -hypersurfaces in with are countably -rectifiable according to Pauls’ definition, even with bi-Lipschitz maps.

Keywords: differential geometry; group theory; measure theory

Free keywords: Carnot groups; codimension-one rectifiability; smooth hypersurface; intrinsic rectifiable set; intrinsic Lipschitz graph

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Ministry reporting: Yes

VIRTA submission year: 2020

JUFO rating: 1