Calculus of Variations and Geometric Measure Theory

A. Figalli - A. Belotto da Silva - A. ParusiƄski - L. Rifford

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

created by figalli on 07 Oct 2022
modified on 10 Feb 2023


Published Paper

Inserted: 7 oct 2022
Last Updated: 10 feb 2023

Journal: Invent. Math.
Year: 2022


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 Hausdor 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 subRiemannian minimizing geodesics, we prove that minimizing sub-Riemannian geodesics in 3-dimensional analytic manifolds are always of class C1, and actually they are analytic outside of a nite set of points.