Accepted Paper
Inserted: 23 may 2024
Last Updated: 14 aug 2025
Journal: Adv. Calc. Var.
Year: 2024
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 \emph{flat chain conjecture}. While previous proofs rely on the delicate task of constructing Lipschitz functions with small $L^\infty$-norm but large derivative along certain directions at all points of a given Lebesgue null set (the so-called \emph{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 $d\omega = \pi$ 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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