A PDE perspective on the flat chain conjecture
Andrea Marchese
TL;DR
The paper surveys progress on the flat chain conjecture, which posits an equivalence between metric currents and flat chains with finite mass in $\mathbb{R}^d$, with complete proofs in the 1D and top-degree cases and open status for intermediate dimensions $1<k<d$. It foregrounds a PDE regularity perspective, linking flat-chain membership to Lipschitz-regularity estimates and decomposability-bundle structure, and showcases a new 1D proof that avoids Alberti representations by leveraging a Poincaré-type framework. A central theme is a Lusin-type regularity program for gradients and $k$-forms, whose validity would imply the full conjecture, alongside a counterexample by Takáč demonstrating limits of stronger conjectures without finite mass. The work highlights how a shift from geometric-width constructions to PDE closability arguments yields powerful insights, with partial progress for higher dimensions and ongoing interest in a Lusin-type theorem for $k$-forms as a path to resolution. Overall, the paper clarifies connections between geometric measure theory and PDE regularity, outlining concrete avenues and obstacles toward a complete proof or refutation of the flat chain conjecture in intermediate dimensions.
Abstract
This survey summarizes recent progress on the flat chain conjecture, which asserts the equivalence between metric currents and flat chains with finite mass in the Euclidean space. In particular, we focus on recent work showing that the conjecture is equivalent to a Lipschitz regularity estimate for a certain PDE.