Singular CR structures of constant Webster curvature and applications

We consider the sphere ${\text{Sph}}^{2n+1}$ equipped with its standard CR structure. In this paper we construct explicit contact forms on $\Sph^{2n+1}\setminus \Sph^{2k+1}$, which are conformal to the standard one and whose related Webster metrics have constant Webster curvature; in particular the curvature is positive if $2k<n-2$. As main applications, we provide two perturbative results. In the first one we prove the existence of infinitely many contact structures on ${\text{Sph}}^{2n+1}\setminus \tau ({\text{Sph}}^1)$ conformal to the standard one and having constant Webster curvature, where $\tau ({\text{Sph}}^1)$ is a small perturbation of ${\text{Sph}}^1$. In the second application, we show that there exist infinitely many bifurcating branches of periodic solutions to the CR Yamabe problem on ${\text{Sph}}^{2n+1}\setminus {\text{Sph}}^1$ having constant Webster curvature.