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We develop a "metrically selfdual" variational calculus for $c$-monotonevector fields between general manifolds $X$ and $Y$, where $c$ is a coupling on$X\times Y$. Remarkably, many of the key properties of classical monotoneoperators known to hold in a linear context, extend to this non-linear setting.This includes an integral representation of $c$-monotone vector fields in termsof $c$-convex selfdual Lagrangians, their characterization as a partial$c$-gradients of antisymmetric Hamiltonians, as well as the property that thesevector fields are generically single-valued. We also use a symmetricMonge-Kantorovich transport to associate to any measurable map its closestpossible $c$-monotone "rearrangement". We also explore how this metricallyselfdual representation can lead to a global variational approach to theproblem of inverting $c$-monotone maps, an approach that has proved efficientfor resolving non-linear equations and evolutions driven by monotone vectorfields in a Hilbertian setting.

Metric Selfduality and Monotone Vector Fields on Manifolds