Uniformization of spherical CR manifolds

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 $$ denote a $CR$ developing map {equation*} Φ:\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 $Φ$ is injective for $n\geq 3$. In the case $n=2$, we also show that $Φ$ is injective under the condition: $s(M)<1$. It then follows that $M$ is uniformizable.

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