In Kähler geometry, Yau’s uniformization conjecture states that any complete noncompact Kähler manifold with positive holomorphic bisectional curvature must be the standard complex Euclidean space. A Sasakian manifold is an odd dimensional counterpart of Kähler geometry. In our recent work, we concern with an CR analogue of Yau’s uniformization conjectures which states that any complete noncompact Sasakian manifold of positive pseudohermitian bisectional curvature must be CR biholomorphic to the standard Heisenberg group.

In this talk, we affirmed the partial results of Yau’s uniformization conjectures on Sasakian manifolds, in particular the CR sharp dimensional estimate and existence of the cone metric at infinity. This is a crucial step toward the CR Yau uniformization conjecture. This is the joint work with Y.-B. Han, C. Lin, and N. Li.