On the codimension of the abnormal set in step two Carnot groups

Abstract

Abstract. In this article we prove that the codimension of the abnormal set of the endpoint map for certain classes of Carnot groups of step 2 is at least three. Our result applies to all step 2 Carnot groups of dimension up to 7 and is a generalisation of a previous analogous result for step 2 free nilpotent groups.

The Sard problem in step 2 and in filiform Carnot groups,

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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 [12] and settles a question emerged in [15]. 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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“…the set of critical values of the endpoint map, is negligible or not; we refer to Section 2 for precise definitions. Despite such a simple formulation, only very partial results are known [1,5,6,16,18] even in settings with a rich structure such as Carnot groups [3,7,9,10,11,12,15]. The goal of this note is to provide a contribution in two meaningful classes of Carnot groups: those with nilpotency step 2, and filiform ones.…”

“…The Sard problem in step 2 Carnot groups has already been answered affirmatively in [12]; however, the question of getting finer estimates on the size of the abnormal set was left open. In fact, in [15] it was conjectured that the abnormal set Abn G in a Carnot group G of step 2 has codimension at least 3. The conjecture is true in free Carnot groups of step 2, as proved in [12], and in some classes of step 2 Carnot groups [15] including all groups of topological dimension up to 7.…”

A dynamical approach to the Sard problem in Carnot groups

“…We believe that, in Theorem 3, the bound 1 on the codimension of Abn G can be improved and we conjecture that it holds with a lower bound 3 (see [30] for an analogous open question in step 2 Carnot groups). We are able to prove our conjecture at least when G is the free Carnot group of rank 3 and step 3.…”

“…A detailed study of the singular set has been carried out in [13,12] for 3-dimensional analytic sub-Riemannian manifolds with 2-dimensional analytic horizontal distributions: it turns out that such a set has Hausdorff dimension 1 and, actually, it is a semi-analytic curve. Other partial or related results are contained in [34,32,26,27,7,30]. Different approaches to study singular curves are found e.g.…”