Differential forms for fractal subspaces and finite energy coordinates

TL;DR

This paper develops a theory of differential forms on fractal subsets of Euclidean space, connecting classical differential geometry with fractal analysis and Dirichlet forms, and demonstrates applications to fractals like the harmonic Sierpinski gasket.

Contribution

It introduces a new framework for differential forms on fractals using pointwise cotangent spaces and links it to Dirichlet form theory, expanding geometric analysis on fractal sets.

Findings

Differential forms form a Banach algebra on fractal subsets.

Forms can be integrated along rectifiable paths on fractals.

The framework applies to the harmonic Sierpinski gasket.

Abstract

This paper introduces a notion of differential forms on closed, potentially fractal, subsets of Euclidean space by defining pointwise cotangent spaces using the restriction of $C^1$ functions to this set. Aspects of cohomology are developed: it is shown that the differential forms are a Banach algebra and it is possible to integrate these forms along rectifiable paths. These definitions are connected to the theory of differential forms on Dirichlet spaces by considering fractals with finite energy coordinates. In this situation, the $C^1$ differential forms project onto the space of Dirichlet differential forms. Further, it is shown that if the intrinsic metric of a Dirichlet form is a length space, then the image of any rectifiable path through a finite energy coordinate sequence is also rectifiable. The example of the harmonic Sierpinski gasket is worked out in detail.