Singular CR structures of constant Webster curvature and applications

We consider the sphere (Formula presented.) equipped with its standard contact form. In this paper, we construct explicit contact forms on (Formula presented.), which are conformal to the standard one and whose related Webster metrics have constant Webster curvature; in particular, it is positive if (Formula presented.). As main applications, we provide two perturbative results. In the first one, we prove the existence of infinitely many contact forms on (Formula presented.) conformal to the standard one and having constant Webster curvature, where (Formula presented.) is a small perturbation of (Formula presented.). In the second application, we show that there exist infinitely many bifurcating branches of periodic solutions to the CR Yamabe problem on (Formula presented.) having constant Webster curvature.