Volume 28, 2022
|Number of page(s)||20|
|Published online||22 December 2022|
Dipartimento di Matematica “T. Levi-Civita”,
via Trieste 63,
2 Département de Mathématiques, Ch. du musée 23, 1700 Fribourg (CH)
*** Corresponding author: firstname.lastname@example.org
Accepted: 6 November 2022
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.
Mathematics Subject Classification: 53C17 / 58K05 / 22E25
Key words: Sard problem / Carnot groups / abnormal curves
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).
© The authors. Published by EDP Sciences, SMAI 2022
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