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

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...

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Bibliographic Details
Published inInventiones mathematicae Vol. 229; no. 1; pp. 395 - 448
Main Authors da Silva, A. Belotto, Figalli, A., Parusiński, A., Rifford, L.
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.07.2022
Springer Nature B.V
Springer Verlag
Subjects
Differential Geometry
Dimensional analysis
Geodesy
Geometry
Manifolds (mathematics)
Mathematical analysis
Mathematics
Mathematics and Statistics
Optimization and Control
Regularity
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Summary: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.
ISSN:0020-9910
1432-1297
DOI:10.1007/s00222-022-01111-2