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dc.contributor.authorAntonelli, Gioacchino
dc.contributor.authorLe Donne, Enrico
dc.date.accessioned2021-06-07T08:19:58Z
dc.date.available2021-06-07T08:19:58Z
dc.date.issued2020
dc.identifier.citationAntonelli, G., & Le Donne, E. (2020). Pauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces. <i>Nonlinear Analysis : Theory, Methods and Applications</i>, <i>200</i>, Article 111983. <a href="https://doi.org/10.1016/j.na.2020.111983" target="_blank">https://doi.org/10.1016/j.na.2020.111983</a>
dc.identifier.otherCONVID_35775322
dc.identifier.urihttps://jyx.jyu.fi/handle/123456789/76297
dc.description.abstractThis 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.en
dc.format.mimetypeapplication/pdf
dc.language.isoeng
dc.publisherElsevier
dc.relation.ispartofseriesNonlinear Analysis : Theory, Methods and Applications
dc.rightsCC BY 4.0
dc.subject.otherCarnot groups
dc.subject.othercodimension-one rectifiability
dc.subject.othersmooth hypersurface
dc.subject.otherintrinsic rectifiable set
dc.subject.otherintrinsic Lipschitz graph
dc.titlePauls rectifiable and purely Pauls unrectifiable smooth hypersurfaces
dc.typearticle
dc.identifier.urnURN:NBN:fi:jyu-202106073513
dc.contributor.laitosMatematiikan ja tilastotieteen laitosfi
dc.contributor.laitosDepartment of Mathematics and Statisticsen
dc.contributor.oppiaineMatematiikkafi
dc.contributor.oppiaineGeometrinen analyysi ja matemaattinen fysiikkafi
dc.contributor.oppiaineAnalyysin ja dynamiikan tutkimuksen huippuyksikköfi
dc.contributor.oppiaineMathematicsen
dc.contributor.oppiaineGeometric Analysis and Mathematical Physicsen
dc.contributor.oppiaineAnalysis and Dynamics Research (Centre of Excellence)en
dc.type.urihttp://purl.org/eprint/type/JournalArticle
dc.type.coarhttp://purl.org/coar/resource_type/c_2df8fbb1
dc.description.reviewstatuspeerReviewed
dc.relation.issn0362-546X
dc.relation.volume200
dc.type.versionpublishedVersion
dc.rights.copyright© 2020 the Authors
dc.rights.accesslevelopenAccessfi
dc.relation.grantnumber713998
dc.relation.grantnumber713998
dc.relation.grantnumber322898
dc.relation.grantnumber288501
dc.relation.projectidinfo:eu-repo/grantAgreement/EC/H2020/713998/EU//GeoMeG
dc.subject.ysomittateoria
dc.subject.ysodifferentiaaligeometria
dc.subject.ysoryhmäteoria
dc.format.contentfulltext
jyx.subject.urihttp://www.yso.fi/onto/yso/p13386
jyx.subject.urihttp://www.yso.fi/onto/yso/p16682
jyx.subject.urihttp://www.yso.fi/onto/yso/p12497
dc.rights.urlhttps://creativecommons.org/licenses/by/4.0/
dc.relation.doi10.1016/j.na.2020.111983
dc.relation.funderEuropean Commissionen
dc.relation.funderResearch Council of Finlanden
dc.relation.funderResearch Council of Finlanden
dc.relation.funderEuroopan komissiofi
dc.relation.funderSuomen Akatemiafi
dc.relation.funderSuomen Akatemiafi
jyx.fundingprogramERC Starting Granten
jyx.fundingprogramAcademy Project, AoFen
jyx.fundingprogramAcademy Research Fellow, AoFen
jyx.fundingprogramERC Starting Grantfi
jyx.fundingprogramAkatemiahanke, SAfi
jyx.fundingprogramAkatemiatutkija, SAfi
jyx.fundinginformationE.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’).
dc.type.okmA1


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