Integrals and Potentials of Differential 1-forms on the Sierpinski Gasket

TL;DR

This paper develops a framework for integrating differential 1-forms on the Sierpinski gasket, enabling potential theory analysis and establishing a de Rham duality theorem for this fractal.

Contribution

It introduces a novel definition of integrals of differential forms on the Sierpinski gasket and proves key theorems like de Rham reconstruction, Hodge decomposition, and duality in this fractal setting.

Findings

Defined integrals of differential 1-forms on the Sierpinski gasket.

Proved de Rham reconstruction and Hodge decomposition for 1-forms.

Established de Rham duality theorem for the fractal K.

Abstract

We provide a definition of integral, along paths in the Sierpinski gasket K, for differential smooth 1-forms associated to the standard Dirichlet form K. We show how this tool can be used to study the potential theory on K. In particular, we prove: i) a de Rham reconstruction of a 1-form from its periods around lacunas in K; ii) a Hodge decomposition of 1-forms with respect to the Hilbertian energy norm; iii) the existence of potentials of smooth 1-forms on a suitable covering space of K. We finally show that this framework provides versions of the de Rham duality theorem for the fractal K.