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

Abstract

A real hypersurface in ${\mathbb{C}}^2$ is said to be Reinhardt if it is invariant under the standard ${\mathbb{T}}^2$-action on ${\mathbb{C}}^2$. 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 ${\mathbb{R}}^2$. 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 ${\mathbb{T}}^2$-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.

11 pages

References in corpus (4)

• One-Dimensional Quasi-Exactly Solvable Schrödinger Equations
• Nonnegativity of the CR Paneitz operator for embeddable CR manifolds
• The sharp upper bounds for the first positive eigenvalue of the Kohn-Laplacian on compact strictly pseudoconvex hypersurfaces
• The Lichnerowicz--Obata theorem for the Kohn Laplacian in three dimensions