Structural properties of one-dimensional metric currents: SBV-representations, connectedness and the flat chain conjecture

Abstract: A comprehensive study of one-dimensional metric currents and their relationship to the geometry of metric spaces is presented. We resolve the one-dimensional flat chain conjecture in this general setting, by proving that its validity is equivalent to a simple geometric connectedness property. More precisely, we prove that metric currents can be approximated in the mass norm by normal currents if and only if every $1$-rectifiable set can be covered by countably many Lipschitz curves up to an $\mathscr{H}^1$-negligible set. Building on this, we demonstrate that any $1$-current in a Banach space can be completed into a cycle by a rectifiable current, with the added mass controlled by the Kantorovich--Rubinstein norm of its boundary. We further refine our approximation result by showing that these currents can be approximated by polyhedral currents modulo a cycle. Finally, in arbitrary complete metric spaces, we establish a Smirnov-type decomposition for one-dimensional currents. This decomposition expresses such currents as a superposition, without mass cancellation, of currents associated with curves of bounded variation that have a vanishing Cantor part.