Inequalities and bounds for the eigenvalues of the sub-Laplacian on a strictly pseudoconvex CR manifold

TL;DR

This paper derives new inequalities and bounds for the eigenvalues of the sub-Laplacian operator on strictly pseudoconvex CR manifolds, extending previous results in the context of the Heisenberg group.

Contribution

It introduces generalized eigenvalue inequalities for the sub-Laplacian on CR manifolds, extending prior work by Niu and Zhang to a broader geometric setting.

Findings

Established inequalities for sub-Laplacian eigenvalues on CR manifolds.

Extended Payne-Pólya-Weinberger and Yang inequalities to CR geometry.

Generalized bounds for eigenvalues beyond the Heisenberg group context.

Abstract

We establish inequalities for the eigenvalues of the sub-Laplace operator associated with a pseudo-Hermitian structure on a strictly pseudoconvex CR manifold. Our inequalities extend those obtained by Niu and Zhang \cite{NiuZhang} for the Dirichlet eigenvalues of the sub-Laplacian on a bounded domain in the Heisenberg group and are in the spirit of the well known Payne-P\'{o}lya-Weinberger and Yang universal inequalities.