Calculus of Variations and Geometric Measure Theory
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A. Marchese - S. Stuvard

On the structure of flat chains modulo $p$

created by marchese on 18 Jul 2016
modified by stuvard on 04 Sep 2017


Published Paper

Inserted: 18 jul 2016
Last Updated: 4 sep 2017

Journal: Advances in Calculus of Variations
Year: 2016
Doi: 10.1515/acv-2016-0040

ArXiv: 1607.05138 PDF


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 $1$.

Keywords: Integral currents mod(p), Flat chains mod(p).


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