Structure of Metric 1-currents: approximation by normal currents and representation results

2025/08/11 by Bate, David, Caputo, Emanuele, Takáč, Jakub +2 · 2 citations
#28A75 #30L99 #49Q15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2508.08017

Abstract

We prove the 1-dimensional flat chain conjecture in any complete and quasiconvex metric space, namely that metric 1-currents can be approximated in mass by normal 1-currents. The proof relies on a new Banach space isomorphism theorem, relating metric 1-currents and their boundaries to the Arens-Eells space. As a by-product, any metric 1-current in a complete and separable metric space can be represented as the integral superposition of oriented 1-rectifiable sets, thus dropping a finite dimensionality condition from previous results of Schioppa [Schioppa Adv. Math. 2016, Schioppa J. Funct. Anal. 2016]. The connection between the flat chain conjecture and the representation result is provided by a structure theorem for metric 1-currents in Banach spaces, showing that any such current can be realised as the restriction to a Borel set of a boundaryless normal 1-current. This generalizes, to any Banach space, the 1-dimensional case of a recent result of Alberti-Marchese in Euclidean spaces [Alberti-Marchese 2023]. The argument of Alberti-Marchese requires the strict polyhedral approximation theorem of Federer for normal 1-currents, which we obtain in Banach spaces.

Citations

• Structural properties of one-dimensional metric currents: SBV-representations, connectedness and the flat chain conjecture
• Quantified compactness in Lipschitz-free spaces of [-1,1]ⁿ
• The superposition principle for local 1-dimensional currents
• Alberti representations, rectifiability of metric spaces and higher integrability of measures satisfying a PDE
• Hyperbolic groups and spherical minimal surfaces
• On the structure of flat chains with finite mass
• Geometric and analytic structures on metric spaces homeomorphic to a manifold
• Decomposition of integral metric currents
• Packing and doubling in metric spaces with curvature bounded above
• Cheeger's differentiation theorem via the multilinear Kakeya inequality
• Higher rank hyperbolicity
• Weak BLD mappings and Hausdorff measure
• Convex geodesic bicombings and hyperbolicity
• Metric Currents and Alberti representations
• Structure of metric cycles and normal one-dimensional currents
• Injective hulls of certain discrete metric spaces and groups
• Compactness for manifolds and integral currents with bounded diameter and volume
• Normal and Integral Currents

Cited by

• A PDE perspective on the flat chain conjecture
• A characterization of snowflakes via rectifiability

Related

• Structural properties of one-dimensional metric currents: SBV-representations, connectedness and the flat chain conjecture
• A simple proof of the 1-dimensional flat chain conjecture
• A PDE perspective on the flat chain conjecture
• Structure of metric cycles and normal one-dimensional currents
• The superposition principle for local 1-dimensional currents