2021/06/25 by Daniele Morbidelli, Morbidelli, Daniele, Séverine Rigot +1
Mathematics · #15A75 #20F18 #43A80 #53C17 #Advanced Topics in Algebra #FOS: Mathematics #Geometric Analysis and Curvature Flows #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2106.13490
openalex publication_date 2021/06/25 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
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.