The CR Yamabe conjecture the case n = 1

Let (M,/) be a compact CR manifold of dimension 2n+1 with a contact form /, and L=(2+2/n)(b+R its associated CR conformal laplacien. The CR Yamabe conjecture states that there is a contact form /; on M conformal to / which has a constant Webster curvature. This problem is equivalent to the existence of a function u such that¶¶D. Jerison and J.M. Lee solved the CR Yamabe problem in the case where nS2 and (M,/) is not locally CR equivalent to the sphere S2n+1 of Cn. In a join work with R. Yacoub, the CR Yamabe problem was solved for the case where (M,/) is locally CR equivalent to the sphere S2n+1 for all n. In the present paper, we study the case n=1, left by D. Jerison and J.M. Lee, which completes the resolution of the CR Yamabe conjecture for all dimensions.