Precisely monotone sets in step-2 rank-3 Carnot algebras

A subset of a Carnot group is said to be precisely monotone if the restriction its characteristic function each integral curve every left-invariant horizontal vector field monotone. Equivalently, set h-convex with complement. Such sets have been introduced and classified in Heisenberg setting by Cheeger Kleiner 2010’s. In present paper, we study wider step-2 groups, equivalently algebras. addition general properties, prove classification terms sublevel h-affine functions rank-3 algebras that can seen as generalization one obtained setting. There however significant difference here it known that, unlike setting, there are on free algebra not half-spaces.