Calculus of Variations and Geometric Measure Theory

F. Boarotto - L. Nalon - D. Vittone

The Sard problem in step 2 and in filiform Carnot groups

created by vittone on 31 Mar 2022



Inserted: 31 mar 2022
Last Updated: 31 mar 2022

Year: 2022

ArXiv: 2203.16360 PDF


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 by Le Donne-Montgomery-Ottazzi-Pansu-Vittone $[$Ann. Inst. H. Poincaré Anal. Non Linéaire 2016$]$ and settles a question raised by Ottazzi-Vittone $[$ESAIM: COCV 2019$]$. 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.