An atomic decomposition of one-dimensional metric currents without boundary

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.