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

Abstract

We give sharp and explicit upper bounds for the first positive eigenvalue ${\lambda }_1({\square }_b)$ 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, ${\lambda }_1({\square }_b)$ has a unique maximum value at the CR sphere.

Final version, appears on Mathematische Zeitschrift

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