Sets with constant normal in Carnot groups: properties and examples

TL;DR

This paper studies sets with constant normal in Carnot groups, establishing their regularity, characterizing their structure, providing counterexamples in specific groups, and proving rectifiability in groups of step 4 or less.

Contribution

It offers new regularity results, characterizations, and examples of constant-normal sets in Carnot groups, advancing understanding of their geometric and measure-theoretic properties.

Findings

Constant-normal sets are regularly open and contractible.

Such sets are unions of translations of semisubgroups.

Counterexamples show pathological behaviors in certain Carnot groups.

Abstract

We analyze subsets of Carnot groups that have intrinsic constant normal, as they appear in the blowup study of sets that have finite sub-Riemannian 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 sub-Riemannian-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 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 sub-Riemannian 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.