The sharp upper bounds for the first positive eigenvalue of the Kohn-Laplacian on compact strictly pseudoconvex hypersurfaces
arXiv:1611.10001 · doi:10.1007/s00209-017-1922-z
Abstract
We give sharp and explicit upper bounds for the first positive eigenvalue $λ_1(\Box_b)$ of the Kohn-Laplacian on compact strictly pseudoconvex hypersurfaces in $\mathbb{C}^{n+1}$ in terms of their defining functions. As an application, we show that in the family of real ellipsoids, $λ_1(\Box_b)$ has a unique maximum value at the CR sphere.
Final version, appears on Mathematische Zeitschrift