An equiregular counterexample to the Sard conjecture for sub-Riemannian endpoint maps

Abstract

Lerario, Rizzi and Tiberio disproved the sub-Riemannian Sard conjecture with a rank-two distribution on $\mathbb R^4$, of degree of nonholonomy five at the origin, whose endpoint map from that point has no regular values. Their distribution is neither equiregular nor real-analytic, and the equiregular case has remained open. We construct an equiregular rank-two distribution on an open set diffeomorphic to $\mathbb R^9$, with small growth vector $(2,3,5,8,9)$, whose endpoint map from the origin has no regular values. This growth vector is maximal and generic for rank-two distributions in dimension nine, so the failure of the Sard property is not tied to any degeneracy of the growth vector. Singular curves of the Lerario--Rizzi--Tiberio-type structure built from the same coefficient lift, inside the strip where our distribution is defined, to singular curves, and the vertical displacement of a horizontal loop is an explicit strip integral. Realising an open set of vertical displacements by singular loops reduces to a graded moment problem, which we solve with pairs of bumps of opposite winding, an anisotropic rescaling, and an inverse function theorem uniform over infinitely many scales. We also state, with proofs deferred to a planned longer version, two results about our distribution: a description of its abnormal dynamics in the cotangent bundle, which explains the example from a dynamical point of view, and the non-constancy of its Tanaka symbol. The latter motivates an equiregular counterexample in dimension twelve with constant Tanaka symbol (constant nilpotentisation), announced here and to be proved in a separate paper. The longer version will also contain obstructions to real-analytic and lower-dimensional versions, and a transversal variant of the construction, which also leads to a modification of the Lerario--Rizzi--Tiberio counterexample.