Metric currents, differentiable structures, and Carnot groups

TL;DR

This paper extends the theory of metric currents to spaces with differentiable structures, especially Carnot groups, establishing new chain rule generalizations and characterizations of currents via vector fields and forms.

Contribution

It generalizes the chain rule for metric currents, characterizes currents in Carnot groups, and links vanishing forms to the behavior of currents along specific sets.

Findings

Metric forms vanishing in Cheeger sense also vanish when paired with certain currents.

Currents of absolutely continuous mass are represented by integration against measurable vector fields.

In Carnot groups, currents satisfy specific annihilation conditions related to the horizontal bundle.

Abstract

We examine the theory of metric currents of Ambrosio and Kirchheim in the setting of spaces admitting differentiable structures in the sense of Cheeger and Keith. We prove that metric forms which vanish in the sense of Cheeger on a set must also vanish when paired with currents concentrated along that set. From this we deduce a generalization of the chain rule, and show that currents of absolutely continuous mass are given by integration against measurable $k$-vector fields. We further prove that if the underlying metric space is a Carnot group with its Carnot-Carath\'eodory distance, then every metric current $T$ satisfies $T\lfloor_{\theta}=0$ and $T\lfloor_{d\theta}=0$, whenever $\theta \in \Omega^{1}(\mathbb{G})$ annihilates the horizontal bundle of $\mathbb G$. Moreover, this condition is necessary and sufficient for a metric current with respect to the Riemannian metric to extend to one with respect to the Carnot-Carath\'eodory metric, provided the current either is locally normal, or has absolutely continuous mass.