Regularity of sets with constant intrinsic normal in a class of Carnot groups

Marchi, Marco

Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 429-455

In this Note, we define a class of stratified Lie groups of arbitrary step (that are called “groups of type $☆$” throughout the paper), and we prove that, in these groups, sets with constant intrinsic normal are vertical halfspaces. As a consequence, the reduced boundary of a set of finite intrinsic perimeter in a group of type $☆$ is rectifiable in the intrinsic sense (De Giorgi’s rectifiability theorem). This result extends the previous one proved by Franchi, Serapioni & Serra Cassano in step 2 groups.

Dans cette note, nous définissons une classe de groupes de Lie stratifiés de pas arbitraire (que nous appelons “groupes de type $☆$” dans cet article), et nous montrons que, dans ces groupes, les ensembles à normale intrinsèque constante sont des hyperplans. En conséquence, la frontière réduite d’un ensemble de périmètre intrinsèque fini dans un groupe de type $☆$ est rectifiable au sens intrinsèque (théorème de rectifiabilité de De Giorgi). Ce résultat étend un résultat précédent prouvé par Franchi, Serapioni & Serra Cassano pour les groupes de pas 2.

Classification : 28A75, 49Q15, 58C35
Keywords: Carnot groups, intrinsic perimeter, intrinsic rectifiability
Mots-clés : Groupes de Carnot, périmètre intrinsèque, rectifiabilité intrinsèque

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