  We prove that the absolute spectral gap of any monotone Markov chain
coincides with its optimal Ollivier-Ricci curvature, where the word `optimal'
refers to the choice of the underlying metric. Moreover, we provide a new
expression in terms of local variations of increasing functions, which has
several practical advantages over the traditional variational formulation using
the Dirichlet form. As an illustration, we explicitly determine the optimal
curvature and spectral gap of the non-conservative exclusion process with
heterogeneous reservoir densities on any network, despite the lack of
reversibility.
