The Sard conjecture on Martinet surfaces

TL;DR

This paper proves the Sard conjecture for certain Martinet surfaces in three-dimensional manifolds, showing that the set of points reachable by singular paths has measure zero under specific smoothness conditions.

Contribution

It establishes the Sard conjecture for smooth Martinet surfaces and extends results to certain singular real-analytic cases using divergence control and resolution of singularities.

Findings

Sard conjecture holds for smooth Martinet surfaces.

The conjecture extends to some singular real-analytic cases.

Methods involve divergence control and resolution of singularities.

Abstract

Given a totally nonholonomic distribution of rank two on a three-dimensional manifold we investigate the size of the set of points that can be reached by singular horizontal paths starting from a same point. In this setting, the Sard conjecture states that that set should be a subset of the so-called Martinet surface of 2-dimensional Hausdorff measure zero. We prove that the conjecture holds in the case where the Martinet surface is smooth. Moreover, we address the case of singular real-analytic Martinet surfaces and show that the result holds true under an assumption of non-transversality of the distribution on the singular set of the Martinet surface. Our methods rely on the control of the divergence of vector fields generating the trace of the distribution on the Martinet surface and some techniques of resolution of singularities.