Strong Sard Conjecture and regularity of singular minimizing geodesics for analytic sub-Riemannian structures in dimension 3

A Belotto da Silva (1) , A Figalli (2) , A Parusiński (3) , L Rifford (3, 4)

1 I2M - Institut de Mathématiques de Marseille
2 D-MATH - Department of Mathematics [ETH Zurich]
3 LJAD - Laboratoire Jean Alexandre Dieudonné
4 McTAO - Mathematics for Control, Transport and Applications

In this paper we prove the strong Sard conjecture for sub-Riemannian structures on 3-dimensional analytic manifolds. More precisely, given a totally nonholonomic analytic distribution of rank 2 on a 3-dimensional analytic manifold, we investigate the size of the set of points that can be reached by singular horizontal paths starting from a given point and prove that it has Hausdorff dimension at most 1. In fact, provided that the lengths of the singular curves under consideration are bounded with respect to a given complete Riemannian metric, we demonstrate that such a set is a semianalytic curve. As a consequence, combining our techniques with recent developments on the regularity of sub-Riemannian minimizing geodesics, we prove that minimizing sub-Riemannian geodesics in 3-dimensional analytic manifolds are always of class C 1 , and actually are analytic outside of a finite set of points.

Domaines: Géométrie différentielle [math.DG]; Optimisation et contrôle [math.OC]

A Belotto da Silva, A Figalli, A Parusiński, L Rifford. Strong Sard Conjecture and regularity of singular minimizing geodesics for analytic sub-Riemannian structures in dimension 3. 2018. ⟨hal-01889705v1⟩