Structure of metric cycles and normal one-dimensional currents
Differential Geometry
2014-01-28 v2 Functional Analysis
Metric Geometry
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.
Cite
@article{arxiv.1303.5667,
title = {Structure of metric cycles and normal one-dimensional currents},
author = {Emanuele Paolini and Eugene Stepanov},
journal= {arXiv preprint arXiv:1303.5667},
year = {2014}
}