Uniformization of spherical CR manifolds

Let M be a closed (compact with no boundary) spherical CR manifold of dimension 2n+1. Let M~ be the universal covering of M. Let Φ denote a CR developing mapΦ:M~→S2n+1 where S²ⁿ⁺¹ is the standard unit sphere in complex n+1-space Cⁿ⁺¹. Suppose that the CR Yamabe invariant of M is positive. Then we show that Φ is injective for n≥3. In the case n=2, we also show that Φ is injective under the condition: s(M)<1 where s(M) means the minimum exponent of the integrability of the Green's function for the CR invariant sublaplacian on M~. It then follows that M is uniformizable.