Calculus of Variations and Geometric Measure Theory

E. Paolini - E. Stepanov

Structure of metric cycles and normal one-dimensional currents

created by stepanov on 01 Aug 2012
modified by paolini on 04 May 2017


Published Paper

Inserted: 1 aug 2012
Last Updated: 4 may 2017

Journal: J. Funct. Anal.
Volume: 264
Number: 6
Pages: 1269--1295
Year: 2013
Doi: 10.1016/j.jfa.2012.12.007

ArXiv: 1303.5667 PDF


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.