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

Abstract

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 $$C^1$$ C 1 , and actually they are analytic outside of a finite set of points.

Subdifferentials and minimizing Sard conjecture in sub-Riemannian geometry

“…The latter case verifies indeed the stronger Sard Conjecture. We refer the reader to [17,29] for a few other specific examples of Carnot groups satisfying the Sard Conjecture and to [3,16,14,15,17,18,29,32,34,37,38] for further details, results and discussions on that conjecture.…”