On the estimate of first positive eigenvalue of a sublaplacian in a pseudohermitian manifold

In this paper, we first obtain a CR version of Yau's gradient estimate for eigenfunc- tions of a sublaplacian. Second, by using CR analogue of Li-Yau's eigenvalue estimate, we are able to obtain a lower bound of the first positive eigenvalue in a pseudohermitian manifold of nonvanishing pseudohermitian torsion and nonpositive lower bound on pseudohermitian Ricci curvature.