Lower bounds for the spectral gap and an extension of the Bonnet-Myers theorem

TL;DR

This paper establishes lower bounds for the spectral gap and spectral bounds on Riemannian manifolds based on Ricci curvature, extending classical theorems and applying to various operators and measures.

Contribution

It provides new Ricci curvature-based lower estimates for spectral bounds and gaps, extending the Bonnet-Myers theorem and applying to weighted manifolds and perturbations.

Findings

Lower bounds for spectral bounds in terms of Ricci curvature.

Extension of the Bonnet-Myers theorem on manifold compactness.

Lower bounds for spectral gaps of Ornstein-Uhlenbeck operators.

Abstract

On a fairly general class of Riemannian manifolds M, we prove lower estimates in terms of the Ricci curvature for the spectral bound (when M has infinite volume) and for the spectral gap (when M has finite volume) for the Laplace-Beltrami operator. As a byproduct of our results we obtain an extension of the Bonnet-Myers theorem on the compactness of the manifold. We also prove lower bounds for the spectral gap for Ornstein-Uhlenbeck type operators on weighted manifolds. As an application we prove lower bounds for the spectral gap of perturbations of some radial measures on R n .