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# Advances in Calculus of Variations

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Volume 11, Issue 3

# On the structure of flat chains modulo p

Andrea Marchese
/ Salvatore Stuvard
Published Online: 2017-04-19 | DOI: https://doi.org/10.1515/acv-2016-0040

## Abstract

In this paper, we prove that every equivalence class in the quotient group of integral 1-currents modulo p in Euclidean space contains an integral current, with quantitative estimates on its mass and the mass of its boundary. Moreover, we show that the validity of this statement for m-dimensional integral currents modulo p implies that the family of $\left(m-1\right)$-dimensional flat chains of the form pT, with T a flat chain, is closed with respect to the flat norm. In particular, we deduce that such closedness property holds for 0-dimensional flat chains, and, using a proposition from The structure of minimizing hypersurfaces mod 4 by Brian White, also for flat chains of codimension 1.

MSC 2010: 49Q15

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## About the article

Revised: 2017-03-20

Accepted: 2017-03-23

Published Online: 2017-04-19

Published in Print: 2018-07-01

Funding Source: FP7 Ideas: European Research Council

Award identifier / Grant number: 306246

Both authors are supported by the ERC-grant RAM “Regularity of Area Minimizing currents”, ID 306246.

Citation Information: Advances in Calculus of Variations, Volume 11, Issue 3, Pages 309–323, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258,

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