Calculus of Variations and Geometric Measure Theory

G. Alberti - A. Marchese

On the structure of flat chains with finite mass

created by marchese on 31 Jan 2022
modified on 20 Mar 2023



Inserted: 31 jan 2022
Last Updated: 20 mar 2023

Year: 2022


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 $\phi:\tilde G\to G$ 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 $\phi$ is surjective, so is also the restriction of $\pi$ to the set of flat chains with finite mass of dimension $0$, $1$, $d-1$, $d$.