Improved spectral gap bounds on positively curved manifolds

TL;DR

This paper introduces new lower bounds for the spectral gap of reversible diffusions on compact manifolds using coupling and analytic methods, based on curvature notions like coarse Ricci and Bakry-Emery, especially when curvature is nonnegative.

Contribution

It provides novel spectral gap bounds derived from curvature concepts, combining coupling and analytic techniques for positively curved manifolds.

Findings

Harmonic mean of curvature bounds the spectral gap when curvature is nonnegative

New lower bounds improve understanding of diffusion behavior on manifolds

Methods applicable to various curvature notions like coarse Ricci and Bakry-Emery

Abstract

A coupling method and an analytic one allow us to prove new lower bounds for the spectral gap of reversible diffusions on compact manifolds. Those bounds are based on the a notion of curvature of the diffusion, like the coarse Ricci curvature or the Bakry--Emery curvature-dimension inequalities. We show that when this curvature is nonnegative, its harmonic mean is a lower bound for the spectral gap.