On the structure of flat chains with finite mass

Abstract

We prove that every flat chain with finite mass in ${\mathbb{R}}^d$ with coefficients in a normed abelian group $G$ is the restriction of a normal $G$-current to a Borel set. We deduce a characterization of real flat chains with finite mass in terms of a pointwise relation between the associated measure and vector field. We also deduce that any codimension-one real flat chain with finite mass can be written as an integral of multiplicity-one rectifiable currents, without loss of mass. Given a Lipschitz homomorphism $\varphi :\tilde{G}\to G$ between two groups, we then study the associated map $\pi$ between flat chains in ${\mathbb{R}}^d$ with coefficients in $\tilde{G}$ and $G$ respectively. In the case $\tilde{G}=\mathbb{R}$ and $G={\mathbb{S}}^1$, we prove that if $\varphi$ is surjective, so is the restriction of $\pi$ to the set of flat chains with finite mass of dimension $0$, $1$, $d-1$, $d$.