A simple proof of the 1-dimensional flat chain conjecture

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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi mathvariant="normal">∞</m:mi> </m:msup> </m:math> {L^{\infty}} -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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>d</m:mi> <m:mo>⁢</m:mo> <m:mi>ω</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mi>π</m:mi> </m:mrow> </m:math> {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.