The CR Yamabe conjecture the case $n=1$

TLDR: For the case n = 1, Jerison and Lee as discussed by the authors solved the CR Yamabe conjecture for all dimensions, which is equivalent to the existence of a function u such that u is locally CR equivalent to a sphere S 2n+1 for all n.

Abstract: Let (M,θ) be a compact CR manifold of dimension 2n+1 with a contact form θ, and L=(2+2/n)Δ b +R its associated CR conformal laplacien. The CR Yamabe conjecture states that there is a contact form &θtilde; on M conformal to θ 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 n≥2 and (M,θ) is not locally CR equivalent to the sphere S 2n+1 of C n . In a join work with R. Yacoub, the CR Yamabe problem was solved for the case where (M,θ) is locally CR equivalent to the sphere S 2n+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.