A1 Journal article (refereed)
Semigenerated Carnot algebras and applications to sub-Riemannian perimeter (2021)

Le Donne, E., & Moisala, T. (2021). Semigenerated Carnot algebras and applications to sub-Riemannian perimeter. Mathematische Zeitschrift, 299(3-4), 2257-2285. https://doi.org/10.1007/s00209-021-02744-4

JYU authors or editors

Publication details

All authors or editors: Le Donne, Enrico; Moisala, Terhi

Journal or series: Mathematische Zeitschrift

ISSN: 0025-5874

eISSN: 1432-1823

Publication year: 2021

Publication date: 17/05/2021

Volume: 299

Issue number: 3-4

Pages range: 2257-2285

Publisher: Springer

Publication country: Germany

Publication language: English

DOI: https://doi.org/10.1007/s00209-021-02744-4

Publication open access: Openly available

Publication channel open access: Partially open access channel

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/75757

Web address of parallel published publication (pre-print): https://arxiv.org/abs/2004.08619

Abstract

This paper contributes to the study of sets of finite intrinsic perimeter in Carnot groups. Our intent is to characterize in which groups the only sets with constant intrinsic normal are the vertical half-spaces. Our viewpoint is algebraic: such a phenomenon happens if and only if the semigroup generated by each horizontal half-space is a vertical half-space. We call semigenerated those Carnot groups with this property. For Carnot groups of nilpotency step 3 we provide a complete characterization of semigeneration in terms of whether such groups do not have any Engel-type quotients. Engel-type groups, which are introduced here, are the minimal (in terms of quotients) counterexamples. In addition, we give some sufficient criteria for semigeneration of Carnot groups of arbitrary step. For doing this, we define a new class of Carnot groups, which we call type (◊)(◊) and which generalizes the previous notion of type (⋆)(⋆) defined by M. Marchi. As an application, we get that in type (◊)(◊) groups and in step 3 groups that do not have any Engel-type algebra as a quotient, one achieves a strong rectifiability result for sets of finite perimeter in the sense of Franchi, Serapioni, and Serra-Cassano.

Keywords: group theory; differential geometry; measure theory

Free keywords: Carnot algebra; horizontal half-space; semigroup generated; Lie wedge; constant intrinsic normal; finite sub-Riemannian perimeter; Engel-type algebras; tipe diamond; trimmed algebra

Contributing organizations

Ministry reporting: Yes

Reporting Year: 2021

JUFO rating: 2