Euclidean flat chain conjecture

Determine whether, for 1 ≤ k ≤ n, every metric k-current in Euclidean space ℝ^n corresponds to a classical k-flat chain; equivalently, show that the space of metric k-currents M_k(ℝ^n) coincides with the space of classical k-flat chains. This asks whether metric k-currents and classical k-flat chains are the same objects in dimensions 1 through n in ℝ^n.

Background

Ambrosio and Kirchheim introduced metric currents to extend the theory of currents beyond smooth settings. A central problem they formulated is whether metric currents agree with classical flat chains in Euclidean space, now known as the flat chain conjecture.

Partial progress is known: the conjecture is resolved for k=1 (in various generalities) and for k=n (via work of De Philippis and Rindler). This paper proves the k=1 case in any complete quasiconvex metric space, but the general Euclidean statement for intermediate dimensions remains open.