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On the structure of flat chains modulo p

Andrea Marchese and Salvatore Stuvard

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 (m-1)-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

Communicated by Frank Duzaar


Funding source: FP7 Ideas: European Research Council

Award Identifier / Grant number: 306246

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

Acknowledgements

The authors would like to thank Camillo De Lellis for having posed the problem and for helpful discussions. We are also grateful to the anonymous referee for his/her valuable comments.

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Received: 2016-08-29
Revised: 2017-03-20
Accepted: 2017-03-23
Published Online: 2017-04-19
Published in Print: 2018-07-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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