Abstract. We give sharp and explicit upper bounds for the first positive eigenvalue λ1( b ) of the Kohn-Laplacian on compact strictly pseudoconvex hypersurfaces in C n+1 in terms of their defining functions. As an application, we show that in the family of real ellipsoids, λ1( b ) has a unique maximum value at the CR sphere.
<abstract><p>Let $ R_{{\cal A}} $ be the Cartan classical domains of type Ⅲ and Ⅳ, and $ \Delta_g $ is assumed to be the Laplace-Beltrami operator associated to the Bergman metric $ g $ on $ R_{{\cal A}} $. In this paper, we derive an estimate for $ \lambda_1(\Delta_g) $, which is the bottom of the spectrum of $ \Delta_g $ on $ R_{{\cal A}} $.</p></abstract>
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