The Webster Scalar Curvature and Sharp Upper and Lower Bounds for the First Positive Eigenvalue of the Kohn-Laplacian on Real Hypersurfaces

Let (M,.) be a compact strictly pseudoconvex pseudohermitian manifold which is CR embedded into a complex space. In an earlier paper, Lin and the authors gave several sharp upper bounds for the first positive eigenvalue .1 of the Kohn-Laplacian .b on (M,.). In the present paper, we give a sharp upper bound for .1, generalizing and extending some previous results. As a corollary, we obtain a Reilly-type estimate when M is embedded into the standard sphere. In another direction, using a Lichnerowicz-type estimate by Chanillo, Chiu, and Yang and an explicit formula for the Webster scalar curvature, we give a lower bound for .1 when the pseudohermitian structure . is volume-normalized.