2016/07/31 by Andrea Marchese, Salvatore Stuvard
Mathematics · #Boundary (topology) #Chain (unit) #Codimension #Combinatorics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematics #Modulo #Physics #Pure mathematics #Quotient #math.AP #math.FA #msc:49Q15
paper · pdf · doi:10.1515/acv-2016-0040
published as Adv. Calc. Var. 11 (2018), no. 3, 309-323 · 19 pages. Final version, to appear in Adv. Calc. Var
openalex created_date 2016/08/23 · arxiv created 2017/03/23 · openalex publication_date 2017/04/19 · arxiv updated 2018/07/12 · openalex updated_date 2026/08/05
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:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>m</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> (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.