Metric Selfduality and Monotone Vector Fields on Manifolds

TL;DR

This paper extends the theory of monotone vector fields to general manifolds using a metrically selfdual variational calculus, enabling new integral representations and a variational approach to invert $c$-monotone maps.

Contribution

It introduces a non-linear, manifold-based framework for $c$-monotone vector fields, generalizing classical monotone operator theory with integral representations and variational methods.

Findings

Many properties of classical monotone operators extend to the non-linear manifold setting.

Provides an integral representation of $c$-monotone vector fields via $c$-convex selfdual Lagrangians.

Develops a variational approach for inverting $c$-monotone maps on manifolds.

Abstract

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