  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.
