Uniformization of spherical CR manifolds

Abstract

Let M be a closed (compact with no boundary) spherical CR manifold of dimension 2n+1. Let (M) over tilde be the universal covering of M. Let Phi denote a CR developing map Phi : (M) over tilde -> 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 Phi is injective for n greater than or similar to 3. In the case n = 2, we also show that 41. 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)over tilde>. It then follows that M is uniformizable. (C) 2014 Elsevier Inc. All rights reserved.