The Webster scalar curvature and sharp upper and lower bounds for the first positive eigenvalue of the Kohn-Laplacian on real hypersurfaces

Let $(M,\theta)$ 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 $\lambda_1$ of the Kohn-Laplacian $\Box_b$ on $(M,\theta)$. In the present paper, we give a sharp upper bound for $\lambda_1$, 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 $\lambda_1$ when the pseudohermitian structure $\theta$ is volume-normalized.