Metric currents, differentiable structures, and Carnot groups (Q2915215)

This article examines the relation between the metric currents [\textit{L. Ambrosio} and \textit{B. Kirchheim}, Acta Math. 185, No. 1, 1--80 (2000; Zbl 0984.49025)] and differentiable structures on metric spaces [\textit{S. Keith}, Adv. Math. 183, No. 2, 271--315 (2004; Zbl 1077.46027)]. One of the main results is a compatibility theorem which states that metric forms that vanish in the sense of differentiable structures also vanish in the sense of currents, provided the current is concentrated where the form is defined. Precisely, let \(\omega = \sum_s \beta_s dg_s^1 \wedge \dots\wedge dg_s^k \) be a compactly supported measurable metric \(k\)-form on a metric space \(X\); this means \(\beta_s\) is a bounded Borel function with compact support, and each \(g_s^k\) is locally Lipschitz. Assuming that \(X\) is equipped with a strong measured differentiable structure in the sense of Keith, let \(Y_\omega\) be the set of points in a coordinate patch \(Y\) at which all functions \(g_s^k\) are differentiable and the equality \(\sum_s \beta_s dg_s^1 \wedge \dots\wedge dg_s^k=0 \) holds in the sense of pointwise derivatives. Then for every metric \(k\) current \(T\) of locally finite mass, we have \(T(\omega)=0\) provided that \(T\) is concentrated on \(Y_\omega\).NEWLINENEWLINEAs an application, the author shows that currents of absolutely continuous mass are given by integration against measurable \(k\)-vector fields. The applications of these ideas to the geometry of Carnot-Carathéodory spaces occupy the second half of the paper. The paper is very carefully written and the content is well organized.