Unrectifiable normal currents in Euclidean spaces

TL;DR

This paper constructs a specific example of a normal current in Euclidean space with support that is purely 2-unrectifiable, providing a negative answer to a longstanding question about the rectifiability of supports of normal currents.

Contribution

It demonstrates the existence of a $k$-dimensional normal current with purely 2-unrectifiable support in Euclidean space, answering a question posed by Frank Morgan in 1984.

Findings

Constructed a $k$-dimensional normal current with purely 2-unrectifiable support.

Showed the support of a normal current cannot be purely 1-unrectifiable.

Established that a $(k+1)$-dimensional normal current can be decomposed into rectifiable currents.

Abstract

We construct in $\mathbb{R}^{k+2}$ a $k$-dimensional simple normal current whose support is purely $2$-unrectifiable. The result is sharp because the support of a normal current cannot be purely $1$-unrectifiable and a $(k+1)$-dimensional normal current can be represented as an integral of $(k+1)$-rectifiable currents. This gives a negative answer to the (revised version) of a question of Frank Morgan (1984).