Journal
INVENTIONES MATHEMATICAE
Volume 229, Issue 1, Pages 395-448Publisher
SPRINGER HEIDELBERG
DOI: 10.1007/s00222-022-01111-2
Keywords
-
Categories
Funding
- European Research Council [721675]
- ANR Project LISA [ANR-17-CE40-0023-03]
- ANR Project SRGI Sub-Riemannian Geometry and Interactions [ANR-15-CE40-0018]
Ask authors/readers for more resources
In this paper, we prove the strong Sard conjecture for sub-Riemannian structures on 3-dimensional analytic manifolds. 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. Moreover, we demonstrate that such a set is a semianalytic curve under the condition that the lengths of the singular curves are bounded with respect to a given complete Riemannian metric. We also establish that sub-Riemannian geodesics in 3-dimensional analytic manifolds are always of class C-1 and analytic outside of a finite set of points, combining our techniques with recent developments on the regularity of sub-Riemannian minimizing geodesics.
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.
Authors
I am an author on this paper
Click your name to claim this paper and add it to your profile.
Reviews
Recommended
No Data Available