In this talk, we first derive the sub-gradient estimate for positive pseudoharmonic functions in a complete pseudohermitian (2n+1)-manifold (M,J,θ). It is served as the CR analogue of Yau's gradient estimate. Secondly, we obtain the Bishop-type sub-Laplacian comparison theorem in a class of complete noncompact pseudohermitian manifolds. Finally we have shown the natural analogue of Liouville-type theorems for the sub-Laplacian in a complete pseudohermitian manifold of vanishing pseudohermitian torsion tensors and nonnegative pseudohermitian Ricci curvature tensors.
This is a jointed work with Ting-Jung Kuo and Jingzhi Tie.