Singular CR structures of constant Webster curvature and applications

TL;DR

This paper constructs explicit contact forms on spheres minus lower-dimensional spheres with constant Webster curvature and explores their applications to the CR Yamabe problem, revealing infinitely many solutions and bifurcations.

Contribution

It introduces explicit conformal contact forms with constant Webster curvature on spheres minus spheres and demonstrates their applications to existence and bifurcation results in the CR Yamabe problem.

Findings

Existence of infinitely many contact structures with constant Webster curvature on spheres minus perturbed spheres.

Identification of infinitely many bifurcating branches of solutions to the CR Yamabe problem.

Abstract

We consider the sphere $\Sph^{2n+1}$ 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 $\Sph^{2n+1}\setminus \tau(\Sph^{1})$ conformal to the standard one and having constant Webster curvature, where $\tau(\Sph^{1})$ is a small perturbation of $\Sph^1$. In the second application, we show that there exist infinitely many bifurcating branches of periodic solutions to the CR Yamabe problem on $\Sph^{2n+1}\setminus \Sph^{1}$ having constant Webster curvature.