Abstract
Sub-Riemannian spaces are spaces whose metric structure may be viewed as a constrained geometry, where motion is only possible along a given set of directions, changing from point to point. The simplest example of such spaces is given by the so-called Heisenberg group. The characteristic constrained motion of sub-Riemannian spaces has numerous applications ranging from robotic control in engineering to neurobiology where it arises naturally in functional magnetic resonance imaging (FMRI). It also arises naturally in other branches of pure mathematics as Cauchy-Riemann geometry, complex hyperbolic spaces, and jet spaces. In this paper, we review the use of the relationship between Heisenberg geometry and Cauchy-Riemann (CR) geometry. More precisely, we focus on the resolution of the Yamabe Conjecture which was definitely solved by techniques related to the theory of critical points at infinity. These techniques were first introduced by A. Bahri and H. Brezis for the Yamabe conjecture in the Riemannian settings. We also review the problem of the prescription of the scalar curvature using the same techniques which were studied first by A. Bahri and J. M. Coron as well as the multiplicity of solutions. Finally, we announce in this direction new existence results for Cauchy-Riemann spheres.