Uniformization of spherical CR manifolds

Abstract

Let $M$ be a closed (compact with no boundary) spherical $CR$ manifold of dimension $2n+1$. Let $\widetilde{M}$ be the universal covering of $M.$ Let $% \Phi $ denote a $CR$ developing map {equation*} \Phi :\widetilde{M}\rightarrow S^{2n+1} {equation*}% where $S^{2n+1}$ is the standard unit sphere in complex $n+1$-space $C^{n+1}$% . Suppose that the $CR$ Yamabe invariant of $M$ is positive. Then we show that $\Phi $ is injective for $n\geq 3$. In the case $n=2$, we also show that $\Phi $ is injective under the condition: $s(M)<1$. It then follows that $M$ is uniformizable.