A simple proof of the 1-dimensional flat chain conjecture

We provide a new proof of the fact that metric 1-currents with compact support in the Euclidean space correspond to Federer–Fleming flat chains; that is, the 1-dimensional case of the so-called flat chain conjecture. While previous proofs rely on the delicate task of constructing Lipschitz functions with small L∞-norm but large derivative along certain directions at all points of a given Lebesgue null set (the so-called width functions), our approach is based primarily on Poincaré’s lemma. This perspective allows us to identify a regularity question concerning the solvability of the equation dω = π for differential k-forms, a question that is closely related to the general validity of the flat chain conjecture in higher dimensions.