The Sard problem in step 2 and in filiform Carnot groups

2022/03/30 by Francesco Boarotto, Boarotto, Francesco, Luca Nalon +3
Mathematics · #22E25 #53C17 #58K05 #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology #Metric Geometry (math.MG) #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2203.16360

openalex publication_date 2022/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Sard problem for the endpoint map in some well-known classes of Carnot groups. Our first main result deals with step 2 Carnot groups, where we provide lower bounds (depending only on the algebra of the group) on the codimension of the abnormal set; it turns out that our bound is always at least 3, which improves the result proved in arXiv:1503.03610 and settles a question emerged in arXiv:1709.02854. In our second main result we characterize the abnormal set in filiform groups and show that it is either a horizontal line, or a 3-dimensional algebraic variety.

Related

• On the codimension of the abnormal set in step two Carnot groups
• A dynamical approach to the Sard problem in Carnot groups
• Sard properties for polynomial maps in infinite dimension
• ODE trajectories as abnormal curves in Carnot groups
• Universal Differentiability Sets in Carnot Groups of Arbitrarily High\n Step