The Sard problem in step 2 and in filiform Carnot groups,
ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 75

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 Le Donne et al. [Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016) 1639–1666] and settles a question emerged in Ottazzi and Vittone [ESAIM: COCV 25 (2019) 18]. 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.

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DOI : 10.1051/cocv/2022074
Classification : 53C17, 58K05, 22E25
Keywords: Sard problem, Carnot groups, abnormal curves 
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     year = {2022},
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Boarotto, Francesco; Nalon, Luca; Vittone, Davide. The Sard problem in step 2 and in filiform Carnot groups,. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 75. doi: 10.1051/cocv/2022074

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Cité par Sources :

L.N. is partially supported by the Swiss National Science Foundation (grant 200021-204501 Regularity of sub-Riemannian geodesics and applications) and by the European Research Council (ERC Starting Grant 713998 GeoMeG Geometry of Metric Groups).

D.V. is supported by University of Padova and by GNAMPA of INdAM (Italy).