In this paper, we first obtain a CR version of Yau’s gradient estimate for eigenfunctions 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.
"On the estimate of the first positive eigenvalue of a sublaplacian in a pseudohermitian manifold." Asian J. Math. 18 (5) 929 - 946, November 2014.