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Sets with constant normal in Carnot groups: properties and examples

Bellettini, C; Le Donne, E; (2021) Sets with constant normal in Carnot groups: properties and examples. Commentarii Mathematici Helvetici , 96 (1) pp. 149-198. 10.4171/CMH/510. Green open access

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Abstract

We analyze subsets of Carnot groups that have intrinsic constant normal, as they appear in the blowup study of sets that have finite subRiemannian perimeter. The purpose of this paper is threefold. First, we prove some mild regularity and structural results in arbitrary Carnot groups. Namely, we show that for every constant-normal set in a Carnot group its subRiemannian-Lebesgue representative is regularly open, contractible, and its topological boundary coincides with the reduced boundary and with the measure-theoretic boundary. We infer these properties from a metric cone property. Such a cone will be a semisubgroup with nonempty interior that is canonically associated with the normal direction. We characterize the constant-normal sets exactly as those that are arbitrary unions of translations of such semisubgroups. Second, making use of such a characterization, we provide some pathological examples in the specific case of the free-Carnot group of step 3 and rank 2. Namely, we construct a constant normal set that, with respect to any Riemannian metric, is not of locally finite perimeter; we also construct an example with non-unique intrinsic blowup at some point, showing that it has different upper and lower subRiemannian density at the origin. Third, we show that in Carnot groups of step 4 or less, every constant-normal set is intrinsically rectifiable, in the sense of Franchi, Serapioni, and Serra Cassano.

Type: Article
Title: Sets with constant normal in Carnot groups: properties and examples
Open access status: An open access version is available from UCL Discovery
DOI: 10.4171/CMH/510
Publisher version: https://doi.org/10.4171/CMH/510
Language: English
Additional information: This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Constant horizontal normal, monotone direction, cone property, semigroup generated, Carnot–Lebesgue representative, Lie wedge, free Carnot group, intrinsic rectifiable set, intrinsic Lipschitz graph, subRiemannian perimeter measure
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/10121124
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