Metric Currents and Alberti representations

Abstract

We relate Ambrosio-Kirchheim metric currents to Alberti representations and Weaver derivations. In particular, given a metric current $T$, we show that if the module $\mathcal{X}(\Vert T\Vert )$ of Weaver derivations is finitely generated, then $T$ can be represented in terms of derivations; this extends previous results of Williams. Applications of this theory include an approximation of $1$-dimensional metric currents in terms of normal currents and the construction of Alberti representations in the directions of vector fields.

arXiv commentExposition improved thanks to referee; 5 more pages