Existence of nonconstant CR-holomorphic functions of polynomial growth in Sasakian manifolds

We show that there exists a nonconstant CR-holomorphic function of polynomial growth in a complete noncompact Sasakian manifold of nonnegative pseudohermitian bisectional curvature with the CR maximal volume growth property. This is the very first step toward the CR analogue of the Yau uniformization conjecture which states that any complete noncompact Sasakian manifold of positive pseudohermitian bisectional curvature is CR biholomorphic to the standard Heisenberg group. More precisely, we first construct CR-holomorphic functions with controlled growth in a sequence of exhaustion domains in Sasakian manifolds by applying the Cheeger–Colding theory. Secondly, via the CR analogue of a tangent cone at infinity and the three-circle theorem, we are able to take the subsequence to obtain a nonconstant CR-holomorphic function of polynomial growth.