On a fairly general class of Riemannian manifolds \(\mathcal {M}\) , we prove lower estimates in terms of the Ricci curvature for the spectral bound (when \(\mathcal {M}\) has infinite volume) and for the spectral gap (when \(\mathcal {M}\) 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 some probability measures \(e^{-V(x)} dx\) with a potential V which is radial only inside or outside a certain ball of \({\mathbb {R}}^n\) .
