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This paper studies, using the Bochner technique, a sharp lower bound of the first eigenvalue of a subelliptic Laplace operator on a strongly pseudoconvex CR manifold in terms of its pseudo-Hermitian geometry. For dimensions greater than or equal to 7, the lower bound under a condition on the Ricci curvature and the torsion was obtained by Greenleaf. We give a proof for all dimensions greater than or equal to 5. For dimension 3, the sharp lower bound is proved under a condition which also involves a distinguished covariant derivative of the torsion.

The sharp lower bound for the first positive eigenvalue of a sub-Laplacian on a pseudo-Hermitian manifold