Examples of $2$-unrectifiable normal currents

TL;DR

This paper constructs new examples of normal currents in metric spaces with purely 2-unrectifiable supports and explores their properties using inverse systems of cube complexes, revealing insights into their geometric and measure-theoretic structure.

Contribution

It introduces novel constructions of normal currents with purely 2-unrectifiable supports and demonstrates their realization as limits of cube complex currents in $l^$.

Findings

Supports are purely 2-unrectifiable with Nagata dimension N

Normal currents can be approximated by cube complex currents in flat distance

Constructs examples with simple associated vector fields

Abstract

We construct new examples of normal (metric) currents using inverse systems of cube complexes. For any $N\ge 2$ we provide examples of $N$-dimensional normal currents whose associated vector fields are simple, and whose supports are purely $2$-unrectifiable and have Nagata dimension $N$. We show that in $l^\infty$ normal currents can be realized as limits in the flat distance of currents associated to cube complexes.