# 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.

## Key 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.