Published Paper

Inserted: 7 oct 2022
Last Updated: 19 aug 2024

Journal: Invent. Math.
Year: 2022

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

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