Rectifiability of flat chains in Banach spaces with coefficients in $Z_p$

TL;DR

This paper refines the understanding of flat chains with coefficients in Z_p in Banach spaces, focusing on rectifiability and closure properties under weaker convergence and mass bounds, extending the theory of currents.

Contribution

It provides a detailed analysis of flat chains mod p, establishing rectifiability and closure results using isoperimetric inequalities and quotient constructions.

Findings

Rectifiability of measure-theoretic support established.

Closure theorem for integer rectifiable currents under flat distance mod p.

Application of isoperimetric inequality with universal constants.

Abstract

Aim of this paper is a finer analysis of the group of flat chains with coefficients in $Z_p$ introduced in a recent paper by Ambrosio-Katz, by taking quotients of the group of integer rectifiable currents, along the lines of the the Ziemer and Federer approach. We investigate the typical questions of the theory of currents, namely rectifiability of the measure-theoretic support and boundary rectifiability. Our main result can also be interpreted as a closure theorem for the class of integer rectifiable currents with respect to a (much) weaker convergence, induced by flat distance mod $p$, and with respect to weaker mass bounds. A crucial tool in many proofs is the isoperimetric inequality proved by Ambrosio-Katz with universal constants.