The theory of currents in Geometric Measure Theory provides a powerful framework for studying generalized oriented surfaces. I will focus on metric currents – a generalization of classical currents introduced by Ambrosio and Kirchheim (after De Giorgi’s ideas) suitable for analysis on spaces without a differential structure. 

In the context of metric currents, the Flat Chain Conjecture (FCC) proposes the equivalence between metric currents and flat chains in Euclidean space. This conjecture remains open in general, with notable exceptions in the 1-dimensional and top-dimensional cases. In a general metric spaces, one can re-formulate the FCC as a strong mass approximation property by normal currents.

I will present a novel functional and optimal transport approach to this formulation of the FCC for 1-dimensional currents in general metric  spaces.  I will discuss how this can be used to prove every metric current can be covered by a current without boundary in a nearly Optimal way with respect to the Kantorovic—Rubinstein norm. Finally, arriving to a general Smirnov-type decomposition for metric 1-currents, which allows one to re-direct the study of such currents to the study of SBV curves (or the study of pieces of Lipschitz curves). 

This is joint work with Guy Bouchitté from Math Toulon.