Flat chain conjecture in Euclidean space (general m)

Determine whether, in Euclidean space R^d, Ambrosio–Kirchheim metric m-currents correspond to classical Federer–Fleming flat m-chains of finite mass for general m, thereby establishing the full flat chain conjecture beyond the cases m = 1 and m = d.

Background

The paper studies the relationship between Ambrosio–Kirchheim metric currents and classical de Rham/Federer–Fleming currents in Euclidean space. The central open problem, known as the flat chain conjecture, asserts an equivalence between metric m-currents and classical flat m-chains of finite mass.

This work provides another short proof of the conjecture in the one-dimensional case m = 1. The zero-codimensional case m = d is known by De Philippis and Rindler. The general case for 2 ≤ m ≤ d−1 is not settled here and remains conjectural in the presentation.