Structure of metric cycles and normal one-dimensional currents

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

Semicontinuity of eigenvalues under intrinsic flat convergence

“…The result is an easier version of the decomposition theorems of normal one-dimensional currents by Paolini and Stepanov [24,25]. The decomposition theorem for integral currents can also be proved using their results as a starting point, but we chose to instead give a simpler argument, which is possible because the currents are integral.…”

“…An important ingredient in the proof is a Poincaré-like inequality, that is quite technical and is shown in Appendix C. As a by-product, we also record a decomposition theorem for one-dimensional integral currents in Appendix A, that is a simpler version of a recent decomposition theorem by Paolini and Stepanov for one-dimensional normal currents [24,25].…”

Three superposition principles: Currents, continuity equations and curves of measures

“…The results of type (C) for De Rham currents have been proven first by S. Smirnov [28] (later several different proofs have been given for partial results of this kind, see e.g. [27] and references therein, and also [16] for an interesting discrete analogue), and for general metric current in [24,26]. In this paper we show in fact that the result on representation of acyclic metric currents provides (A) for general metric spaces.…”

Metric currents and Alberti representations

“…This method relies on results of [PS12, PS13] on the structure of 1-dimensional normal currents. We state the Paolini-Stepanov decomposition of normal currents using parametrized curves: note, however, that in [PS13] the result is stated using non-parametrized curves. Recall also that the metric space X is assumed Polish.…”