The Neumann problem on the Clifford torus in ${\mathbb{S}}^3$

We discuss the solution of the Neumann problem associated with the CR Yamabe operator on a subset $\mathrm{\Omega }$ of the CR manifold ${\mathbb{S}}^3$ bounded by the Clifford torus $\mathrm{\Sigma }$. We also discuss the Yamabe-type problem of finding a contact form on $\mathrm{\Omega }$ which has zero Tanaka–Webster scalar curvature and for which $\mathrm{\Sigma }$ has constant $p$-mean curvature.