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

TL;DR

This paper establishes sharp upper and lower bounds for the first positive eigenvalue of the Kohn-Laplacian on compact CR manifolds, extending previous results and providing new geometric estimates related to Webster scalar curvature.

Contribution

It generalizes and extends existing bounds for the eigenvalue of the Kohn-Laplacian on CR manifolds, including sharp bounds and Reilly-type estimates.

Findings

Derived a sharp upper bound for the first positive eigenvalue of the Kohn-Laplacian.

Established a Reilly-type estimate for CR embeddings into the sphere.

Provided a lower bound for the eigenvalue using a Lichnerowicz-type estimate and scalar curvature formula.

Abstract

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.