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

TL;DR

This paper establishes a sharp upper bound for the first positive eigenvalue of the Kohn Laplacian on certain Reinhardt hypersurfaces in complex space, linking it to the curvature of a generating curve, with equality only for circular curves.

Contribution

It provides the first explicit upper bound for the eigenvalue in terms of the generating curve's curvature, characterizing the extremal case as a circle.

Findings

The upper bound is sharp and attained only for circular generating curves.

The eigenvalue bound is expressed via the $L^2$-norm of the curvature function.

The result applies to compact, strictly pseudoconvex Reinhardt hypersurfaces with free $ ext{T}^2$-action.

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.