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

TL;DR

This paper investigates the structure of one-dimensional metric currents, proving the flat chain conjecture in general metric spaces by linking it to geometric connectedness, and introduces new approximation and decomposition results.

Contribution

It proves the flat chain conjecture in metric spaces, linking it to connectedness, and develops new approximation and decomposition techniques for metric currents.

Findings

Flat chain conjecture is valid iff 1-rectifiable sets are covered by Lipschitz curves.

Any 1-current in a Banach space can be completed into a cycle with controlled mass.

Established a Smirnov-type decomposition for 1-currents in metric spaces.

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.