A PDE perspective on the flat chain conjecture

TL;DR

This paper reviews recent advances on the flat chain conjecture, highlighting its equivalence to a PDE regularity problem and summarizing key progress in the field.

Contribution

It connects the flat chain conjecture to a specific PDE regularity estimate, providing a new perspective and summarizing recent developments.

Findings

The flat chain conjecture is equivalent to a Lipschitz regularity estimate for a PDE.

Recent work has made significant progress towards proving the conjecture.

The survey consolidates various approaches and results in the field.

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.