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I will introduce the basic notion of pseudhermitian manifold first and derive the sub-gradient estimate for positive pseudoharmonic functions in a complete pseudohermitian (2n + 1)-manifold (M, J, θ) which satisfies the CR sub-Laplacian comparison property. 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 will show the natural analogue of Liouville-type theo- rems for the sub-Laplacian in a complete pseudohermitian manifold of vanishing pseudohermitian torsion tensors and nonnegative pseudohermitian Ricci curvature tensors. This a joint project with Shu-Cheng Chang and Ting-Jung Kuo of National Taiwan University.