Sub-Laplacian eigenvalue bounds on sub-Riemannian manifolds

TL;DR

This paper establishes asymptotically sharp upper bounds for eigenvalues of sub-Laplacians on regular sub-Riemannian manifolds, extending classical Riemannian results to the sub-Riemannian setting.

Contribution

It provides new eigenvalue bounds for sub-Laplacians on conformal sub-Riemannian manifolds, including Sasakian and contact metric cases, generalizing known Riemannian inequalities.

Findings

Eigenvalue bounds are asymptotically sharp as k→∞.

Derived inequalities for Sasakian manifolds with Ricci curvature bounds.

Extended classical Riemannian eigenvalue results to sub-Riemannian geometry.

Abstract

We study eigenvalue problems for intrinsic sub-Laplacians on regular sub-Riemannian manifolds. We prove upper bounds for sub-Laplacian eigenvalues $\lambda_k$ of conformal sub-Riemannian metrics that are asymptotically sharp as $k\to +\infty$. For Sasakian manifolds with a lower Ricci curvature bound, and more generally, for contact metric manifolds conformal to such Sasakian manifolds, we obtain eigenvalue inequalities that can be viewed as versions of the classical results by Korevaar and Buser in Riemannian geometry.