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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 , and actually are analytic outside of a finite set of points.
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Submitted on : Friday, October 12, 2018 - 8:58:12 AM
Last modification on : Sunday, May 1, 2022 - 3:14:36 AM
Long-term archiving on: : Sunday, January 13, 2019 - 1:17:58 PM

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Andre Belotto da Silva, Alessio Figalli, Adam Parusiński, Ludovic Rifford. Strong Sard Conjecture and regularity of singular minimizing geodesics for analytic sub-Riemannian structures in dimension 3. Inventiones Mathematicae, Springer Verlag, 2022, ⟨10.1007/s00222-022-01111-2⟩. ⟨hal-01889705v2⟩

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