The sharp upper bounds for the first positive eigenvalue of the Kohn-Laplacian on compact strictly pseudoconvex hypersurfaces

摘要

We give sharp and explicit upper bounds for the first positive eigenvalue$\lambda_1(\Box_b)$ of the Kohn-Laplacian on compact strictly pseudoconvexhypersurfaces in $\mathbb{C}^{n+1}$ in terms of their defining functions. As anapplication, we show that in the family of real ellipsoids, $\lambda_1(\Box_b)$has a unique maximum value at the CR sphere.