Structure of metric cycles and normal one-dimensional currents

TL;DR

This paper demonstrates that one-dimensional normal currents in metric spaces can be decomposed into integrals of simpler currents, generalizing classical Euclidean results and extending to general metric spaces under certain assumptions.

Contribution

It introduces a natural representation of one-dimensional normal currents as integrals of Lipschitz curve-based currents, extending Smirnov's Euclidean cycle decomposition to metric spaces.

Findings

Representation of currents as integrals of Lipschitz curves

Cycle decomposition into elementary solenoids

Generalization to complete metric spaces

Abstract

We prove that every one-dimensional real Ambrosio-Kirchheim normal current in a Polish (i.e. complete separable metric) space can be naturally represented as an integral of simpler currents associated to Lipschitz curves. As a consequence a representation of every such current with zero boundary (i.e. a cycle) as an integral of so-called elementary solenoids (which are, very roughly speaking, more or less the same as asymptotic cycles introduced by S. Schwartzman)is obtained. The latter result on cycles is in fact a generalization of the analogous result proven by S. Smirnov for classical Whitney currents in a Euclidean space. The same results are true for every complete metric space under suitable set-theoretic assumptions.