AN UPPER BOUND FOR THE FIRST POSITIVE EIGENVALUE OF THE KOHN LAPLACIAN ON REINHARDT REAL HYPERSURFACES

A real hypersurface in C-2 is said to be Reinhardt if it is invariant under the standard T-2-action on 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. 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-action being free). Our bound is expressed in terms of the L-2-norm of the curvature function of the generating curve and is attained if and only if the curve is a circle.