The sharp upper bounds for the first positive eigenvalue of the Kohn-Laplacian on compact strictly pseudoconvex hypersurfaces

TL;DR

This paper establishes precise upper bounds for the first positive eigenvalue of the Kohn-Laplacian on compact strictly pseudoconvex hypersurfaces, revealing the CR sphere as the unique maximizer within ellipsoids.

Contribution

It provides sharp, explicit bounds for the eigenvalue on pseudoconvex hypersurfaces and identifies the CR sphere as the unique maximizer among ellipsoids.

Findings

The first positive eigenvalue has a sharp upper bound in terms of defining functions.

The CR sphere uniquely maximizes the eigenvalue among ellipsoids.

Explicit bounds are derived for hypersurfaces in complex space.

Abstract

We give sharp and explicit upper bounds for the first positive eigenvalue $\lambda_1(\Box_b)$ of the Kohn-Laplacian on compact strictly pseudoconvex hypersurfaces in $\mathbb{C}^{n+1}$ in terms of their defining functions. As an application, we show that in the family of real ellipsoids, $\lambda_1(\Box_b)$ has a unique maximum value at the CR sphere.