Singular CR structures of constant Webster curvature and applications

Abstract

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