Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces
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
DisciplineMatematiikkaGeometrinen analyysi ja matemaattinen fysiikkaAnalyysin ja dynamiikan tutkimuksen huippuyksikköMathematicsGeometric Analysis and Mathematical PhysicsAnalysis and Dynamics Research (Centre of Excellence)
© 2020 the Authors
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. ...
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Related funder(s)European Commission; Academy of Finland
Funding program(s)Academy Project, AoF; Academy Research Fellow, 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 fundingE.L.D. was partially supported by the Academy of Finland (grant 288501 ‘Geometry of subRiemannian groups’ and by grant 322898 ‘Sub-Riemannian Geometry via Metric-geometry and Lie-group Theory’) and by the European Research Council(ERC Starting Grant 713998 GeoMeG ‘Geometry of Metric Groups’).
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