Uniformization of spherical CR manifolds

Let M be a closed (compact with no boundary) spherical CR   manifold of dimension 2n+12n+1. Let M˜ be the universal covering of M. Let Φ denote a CR developing mapΦ:M˜→S2n+1 where S2n+1S2n+1 is the standard unit sphere in complex n+1n+1-space Cn+1Cn+1. Suppose that the CR Yamabe invariant of M is positive. Then we show that Φ   is injective for n⩾3n⩾3. In the case n=2n=2, we also show that Φ   is injective under the condition: s(M)<1s(M)<1 where s(M)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.