Résume | Given a totally nonholonomic distribution of rank two $\Delta$ on a three-dimensional manifold $M$, it is natural to investigate the size of the set of points $\mathcal{X}^x$ that can be reached by singular horizontal paths starting from a same point $x \in M$. In this setting, the Sard conjecture states that $\mathcal{X}^x$ should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero.
I will present a reformulation of the conjecture in terms of the behavior of a singular foliation. By exploring this geometrical framework, in a recent work in collaboration with A. Figalli, L. Rifford and A. Parusinski, we show that the strong version of the conjecture holds for three dimensional analytic varieties, that is, the set $\mathcal{X}^x$ is a countable union of semi-analytic curves. Next, by studying the regularity of the solutions of the set $\mathcal{X}^x$, we show that sub-Riemannian geodesics are all $C^1$. Our methods rely on resolution of singularities of surfaces, vector-fields and metrics; regularity analysis of Poincaré transition maps; and on a symplectic argument, concerning a transversal metric of an isotropic singular foliation. |