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

TL;DR

The paper develops lower bounds for the spectral bottom $s(\Delta)$ on infinite-volume manifolds and the spectral gap $\lambda_1(\Delta)$ on finite-volume manifolds by linking these spectral quantities to averaged Ricci curvature via a mean condition on the smallest eigenvalue of the Ricci tensor. The authors introduce a ball-average quantity $δ(R)$ and employ the Riesz transform to intertwine the Laplacian with Schrödinger-type operators, yielding robust, geometry-driven bounds that survive under form-bounded negative parts and in weighted settings. They extend classical results by proving Bonnet–Myers–type compactness criteria under sub-exponential volume growth and averaged curvature conditions, and they generalize the framework to weighted manifolds and Ornstein–Uhlenbeck type operators. The methods are then applied to perturbations of radial measures on $\mathbb{R}^n$, yielding dimension-free lower bounds for spectral gaps that match known radial-measure results up to constants and illustrate the broad applicability of the approach to geometric-analytic problems.

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 .

Theorems & Definitions (35)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 3.1
• proof
• Lemma 3.2