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

TL;DR

This paper extends the classification of precisely monotone sets from the Heisenberg group to all step-2 rank-3 Carnot algebras, revealing new types of sublevel sets beyond half-spaces.

Contribution

It provides a classification of precisely monotone sets in step-2 rank-3 Carnot algebras, generalizing previous Heisenberg results and identifying novel sublevel set structures.

Findings

Classification of precisely monotone sets in step-2 rank-3 Carnot algebras.

Identification of sublevel sets that differ from traditional half-spaces.

Extension of known results from the Heisenberg setting to a broader class of Carnot groups.

Abstract

A subset of a Carnot group is said to be precisely monotone if the restriction of its characteristic function to each integral curve of every left-invariant horizontal vector field is monotone. Equivalently, a precisely monotone set is a h-convex set with h-convex complement. Such sets have been introduced and classified in the Heisenberg setting by Cheeger and Kleiner in the 2010's. In the present paper, we study precisely monotone sets in the wider setting of step-2 Carnot groups, equivalently step-2 Carnot algebras. In addition to general properties, we prove a classification in step-2 rank-3 Carnot algebras that generalizes the classification already known in the Heisenberg setting using sublevel sets of h-affine functions. A significant novelty is that such sublevel sets can be different from half-spaces.