Densely defined non-closable curl on carpet-like metric measure spaces

TL;DR

This paper investigates the behavior of the exterior derivative operator on certain fractal-like metric spaces, showing it is not closable in specific non-similar Sierpinski carpets with positive measure.

Contribution

It demonstrates the non-closability of the curl operator on non-self-similar Sierpinski carpets and extends this result to a broader class of fractal spaces with higher martingale dimension.

Findings

Curl operator is not closable on these spaces.

Adjoint operator has a trivial domain.

Results apply to spaces structurally similar to Sierpinski carpets.

Abstract

The paper deals with the possibly degenerate behaviour of the exterior derivative operator defined on $1$-forms on metric measure spaces. The main examples we consider are the non self-similar Sierpinski carpets recently introduced by Mackay, Tyson and Wildrick. Although topologically one-dimensional, they may have positive two-dimensional Lebesgue measure and carry nontrivial $2$-forms. We prove that in this case the curl operator (and therefore also the exterior derivative on $1$-forms) is not closable, and that its adjoint operator has a trivial domain. We also formulate a similar more abstract result. It states that for spaces that are, in a certain way, structurally similar to Sierpinski carpets, the exterior derivative operator taking $1$-forms into $2$-forms cannot be closable if the martingale dimension is larger than one.