Gradient estimate of positive eigenfunctions of sub-Laplacian on complete pseudo-Hermitian manifolds

Abstract In this paper, we derive an explicit gradient estimate of positive solutions of Δ b u = − λ u on complete noncompact pseudo-Hermitian manifolds with bounded geometric conditions. As a consequence, we obtain an estimate of the greatest lower bound for the L 2 -spectrum of the sub-Laplacian operator. Moreover, we recapture the Liouville theorem of positive pseudo-harmonic functions on complete noncompact Sasakian manifolds with nonnegative pseudo-Hermitian Ricci curvature.