Bounds for the first eigenvalue of the horizontal Laplacian in positively curved sub-Riemannian manifolds

TL;DR

This paper derives lower bounds for the first eigenvalue of the horizontal Laplacian in step 2 sub-Riemannian manifolds with positive curvature, applicable even with significant torsion.

Contribution

It introduces general methods to estimate eigenvalues in positively curved sub-Riemannian manifolds, accommodating torsion effects.

Findings

Established lower bounds for the first eigenvalue

Applicable to manifolds with significant torsion

Generalized previous results to broader settings

Abstract

We establish lower bounds for the first non-zero eigenvalue for the natural geometric sub-elliptic Laplacian operator defined on sub-Riemannian manifolds of step 2 that satisfy a positive curvature condition. The methods are very general and can be applied even when the sub-Riemannian geometry has considerable torsion.