The first positive eigenvalue of the sub-laplacian on cr spheres

TL;DR

This paper proves that the normalized first positive eigenvalue of the sub-Laplacian on CR spheres reaches its maximum under the standard contact form, establishing an extremal property in CR geometry.

Contribution

It establishes the maximality of the first positive eigenvalue of the sub-Laplacian on CR spheres when using the standard contact form, a new extremal result in CR geometry.

Findings

Maximum normalized eigenvalue achieved by standard contact form

Characterization of extremal eigenvalues on CR spheres

Advancement in understanding spectral geometry of CR structures

Abstract

We prove that the first positive eigenvalue, normalized by the volume, of the sub-Laplacian associated with a strictly pseudoconvex pseudo-Hermitian structure $\\theta$ on the CR sphere S 2n+1 $\subset$ C n+1 , achieves its maximum when $\\theta$ is the standard contact form.