Sets with constant normal in Carnot groups: properties and examples

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