On the Sasakian Structure of Manifolds with Nonnegative Transverse Bisectional Curvature

In this paper, we concern with the Sasaki analogue of Yau uniformization conjecture in a complete noncompact Sasakian manifold with nonnegative transverse bisectional curvature. As a consequence, we confirm that any $5$-dimensional complete noncompact Sasakian manifold with positive transverse bisectional curvature and the maximal volume growth must be CR-biholomorphic to the standard Heisenberg group ${\mathbb{H}}_2$ which can be stated as the standard contact Euclidean $5$-space ${\mathbb{R}}^5$.