An atomic decomposition of one-dimensional metric currents without boundary

TL;DR

This paper establishes an atomic decomposition for 1-dimensional metric currents without boundary, representing them as sums of closed Lipschitz curves with controlled Morrey norms, refining approximation methods in Euclidean space.

Contribution

It introduces a geometric construction for decomposing piecewise-geodesic closed curves into simpler components with length and norm bounds, advancing the understanding of metric currents.

Findings

Atomic decomposition of 1-dimensional metric currents without boundary.

Approximation of divergence-free measures by sums of closed polygonal paths.

Universal bounds on Morrey norms for decomposed curves.

Abstract

This paper proves an atomic decomposition of the space of $1$-dimensional metric currents without boundary, in which the atoms are specified by closed Lipschitz curves with uniform control on their Morrey norms. Our argument relies on a geometric construction which states that for any $\epsilon>0$ one can express a piecewise-geodesic closed curve as the sum of piecewise-geodesic closed curves whose total length is at most $(1+\epsilon)$ times the original length and whose Morrey norms are each bounded by a universal constant times $\epsilon^{-2}$. In Euclidean space, our results refine the state of the art, providing an approximation of divergence free measures by limits of sums of closed polygonal paths whose total length can be made arbitrarily close to the norm of the approximated measure.