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September 5, 2025Open Access

A simple proof of the 1-dimensional flat chain conjecture

AMAndrea MarcheseUniversity of Verona AMAndrea MerloUniversity of Pisa

Key Points

The new proof establishes that metric 1-currents correspond to flat chains, offering a simpler methodology compared to previous constructs.
Key evidence includes the identification of a regularity issue for differential k-forms, essential for understanding flat chain conjectures.
The method relies on Poincaré's lemma, simplifying the construction of Lipschitz functions previously required in the field.
This work may pave the way for validating the flat chain conjecture in higher dimensions, advancing geometric measure theory.

Abstract

Abstract 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 ∞ 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 d ⁢ ω = π d= for differential k -forms, a question that is closely related to the general validity of the flat chain conjecture in higher dimensions.

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