Structure of Metric $1$-currents: approximation by normal currents and representation results

TL;DR

This paper proves the 1-dimensional flat chain conjecture in metric spaces, showing metric 1-currents can be approximated by normal currents and represented as superpositions of rectifiable sets, extending previous results.

Contribution

It introduces a new Banach space isomorphism theorem linking metric 1-currents to the Arens-Eells space and generalizes representation results to all Banach spaces.

Findings

Metric 1-currents can be approximated in mass by normal 1-currents.

Any metric 1-current can be represented as an integral superposition of rectifiable sets.

Established a strict polyhedral approximation theorem for Banach spaces.

Abstract

We prove the $1$-dimensional flat chain conjecture in any complete and quasiconvex metric space, namely that metric $1$-currents can be approximated in mass by normal $1$-currents. The proof relies on a new Banach space isomorphism theorem, relating metric $1$-currents and their boundaries to the Arens-Eells space. As a by-product, any metric $1$-current in a complete and separable metric space can be represented as the integral superposition of oriented $1$-rectifiable sets, thus dropping a finite dimensionality condition from previous results of Schioppa [Schioppa Adv. Math. 2016, Schioppa J. Funct. Anal. 2016]. The connection between the flat chain conjecture and the representation result is provided by a structure theorem for metric $1$-currents in Banach spaces, showing that any such current can be realised as the restriction to a Borel set of a boundaryless normal $1$-current. This generalizes, to any Banach space, the $1$-dimensional case of a recent result of Alberti-Marchese in Euclidean spaces [Alberti-Marchese 2023]. The argument of Alberti-Marchese requires the strict polyhedral approximation theorem of Federer for normal $1$-currents, which we obtain in Banach spaces.