Decomposition of integral metric currents

TL;DR

This paper proves that integral currents in complete metric spaces can be decomposed into indecomposable parts, extending Federer's characterization to metric spaces and exploring applications of these results.

Contribution

It introduces a decomposition theorem for integral currents in metric spaces and characterizes indecomposables in the one-dimensional case, extending classical results.

Findings

Integral currents decompose into indecomposables in complete metric spaces.

Indecomposable 1D currents correspond to injective Lipschitz curves or loops.

Applications of the decomposition are discussed.

Abstract

In the setting of complete metric spaces, we prove that integral currents can be decomposed as a sum of indecomposable components. In the special case of one-dimensional integral currents, we also show that the indecomposable ones are exactly those associated with injective Lipschitz curves or injective Lipschitz loops, therefore extending Federer's characterisation to metric spaces. Moreover, some applications of our main results will be discussed.