A harmonic mean bound for the spectral gap of the Laplacian on Riemannian manifolds
According to the author, most known lower bounds on the spectral gap of the Laplacian using Ricci curvature are based on the infimum of the Ricci curvature, so they can be poor when the Ricci curvature is large everywhere but on a small subset on which it is small. He, however, shows that the harmonic mean of the Ricci curvature also is a lower bound on the spectral gap of the Laplacian, if the Ricci curvature is everywhere nonnegative. In fact, his main theorem states as follows: Let \(M\) be a compact connected Riemannian manifold without boundary, with nonnegative Ricci curvature. Then the following inequality holds: \[ \frac{1}{\lambda_1}\leq\int_M\frac{d\mu(x)}{\kappa(x)} \] where \(\mu\) is the renormalized volume measure \((d\mu =\frac{\,\text{vol}}{\text{vol}(M)})\), and \(\kappa(x)=\inf_{v\in\text{T}_x M,\,\| v\| =1}\text{Ric}(v,v)\). In the last section, the author generalizes this result for a generator \(L\) of a Markovian process under the condition that \(L\) satisfies a certain so-called curvature-dimension inequality.
scientific article; zbMATH DE number 5835325