# Weak KAM theory in higher-dimensional holonomic measure flows

## TL;DR

This paper develops a weak KAM theory for higher-dimensional holonomic measure flows, connecting it to Hamilton-Jacobi equations and providing new characterizations of minimizable Lagrangians.

## Contribution

It introduces a novel weak KAM framework for parameterized cobordisms and holonomic measures, extending classical theory to higher dimensions.

## Key findings

- Existence of weak KAM solutions in the context of holonomic measures
- Characterization of minimizable Lagrangians within this framework
- Development of abstract weak KAM machinery for higher-dimensional flows

## Abstract

We construct a weak KAM theory for parameterized cobordisms and their relaxation, holonomic measures. We find a weak kam solution in that context, and we show that in many cases it corresponds to an exact form that satisfies a version of the Hamilton-Jacobi equation. Along the way, we give a characterization of minimizable Lagrangians, as well as some abstract weak KAM machinery.