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# Advances in Calculus of Variations

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# 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.

Keywords:

MSC 2010: 49Q15

## References

• [1]

Ambrosio L. and Ghiraldin F., Flat chains of finite size in metric spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 30 (2013), no. 1, 79–100. Google Scholar

• [2]

Ambrosio L. and Katz M. G., Flat currents modulo p in metric spaces and filling radius inequalities, Comment. Math. Helv. 86 (2011), no. 3, 557–592. Google Scholar

• [3]

Ambrosio L. and Kirchheim B., Currents in metric spaces, Acta Math. 185 (2000), no. 1, 1–80. Google Scholar

• [4]

Ambrosio L. and Wenger S., Rectifiability of flat chains in Banach spaces with coefficients in ${ℤ}_{p}$, Math. Z. 268 (2011), no. 1–2, 477–506. Google Scholar

• [5]

Brothers J. E., Some open problems in geometric measure theory and its applications suggested by participants of the 1984 AMS Summer Institute, Geometric Measure Theory and the Calculus of Variations (Arcata 1984), Proc. Sympos. Pure Math. 44, American Mathematical Society, Providence (1986), 441–464. Google Scholar

• [6]

De Pauw T. and Hardt R., Rectifiable and flat G chains in a metric space, Amer. J. Math. 134 (2012), no. 1, 1–69. Google Scholar

• [7]

Federer H., Geometric Measure Theory, Grundlehren Math. Wiss. 153, Springer, New York, 1969. Google Scholar

• [8]

Federer H. and Fleming W. H., Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520. Google Scholar

• [9]

Krantz S. G. and Parks H. R., Geometric Integration Theory, Birkhäuser, Boston, 2008. Google Scholar

• [10]

Morgan F., A regularity theorem for minimizing hypersurfaces modulo ν, Trans. Amer. Math. Soc. 297 (1986), no. 1, 243–253. Google Scholar

• [11]

Morgan F., Geometric Measure Theory. A Beginner’s Guide, 4th ed., Elsevier, Amsterdam, 2009. Google Scholar

• [12]

Simon L., Lectures on Geometric Measure Theory, Australian National University, Canberra, 1983. Google Scholar

• [13]

White B., The structure of minimizing hypersurfaces mod 4, Invent. Math. 53 (1979), no. 1, 45–58. Google Scholar

• [14]

White B., A regularity theorem for minimizing hypersurfaces modulo p, Geometric Measure Theory and the Calculus Of Variations (Arcata 1984), Proc. Sympos. Pure Math. 44, American Mathematical Society, Providence (1986), 413–427. Google Scholar

• [15]

White B., Rectifiability of flat chains, Ann. of Math. (2) 150 (1999), no. 1, 165–184. Google Scholar

• [16]

Ziemer W. P., Integral currents $\mathrm{mod}$ 2, Trans. Amer. Math. Soc. 105 (1962), 496–524. Google Scholar

Revised: 2017-03-20

Accepted: 2017-03-23

Published Online: 2017-04-19

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, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258,

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