An upper bound for the first positive eigenvalue of the Kohn Laplacian on Reinhardt real hypersurfaces

Abstract

A real hypersurface in C 2 \mathbb {C}^2 is said to be Reinhardt if it is invariant under the standard T 2 \mathbb {T}^2 -action on C 2 \mathbb {C}^2 . Its CR geometry can be described in terms of the curvature function of its “generating curve”, i.e., the logarithmic image of the hypersurface in the plane R 2 \mathbb {R}^2 . We give a sharp upper bound for the first positive eigenvalue of the Kohn Laplacian associated to a natural pseudohermitian structure on a compact and strictly pseudoconvex Reinhardt real hypersurface having closed generating curve (which amounts to the T 2 \mathbb {T}^2 -action being free). Our bound is expressed in terms of the L 2 L^2 -norm of the curvature function of the generating curve and is attained if and only if the curve is a circle.