Hodge-de Rham Theory on Higher-Dimensional Level-L Sierpinski Gaskets

TL;DR

This paper generalizes Hodge-de Rham theory to higher-dimensional level-l Sierpinski gaskets, constructing differential forms, Laplacians, and harmonic bases on these fractals, with specific results on 1-forms and 2-forms.

Contribution

It extends Hodge-de Rham theory to higher-dimensional fractals, defining forms and harmonic structures on complex Sierpinski gaskets, and analyzes properties of 2-forms under measure assumptions.

Findings

Harmonic extension of 1-forms is established.

A basis for harmonic 1-forms is constructed.

Properties of 2-forms are analyzed under measure assumptions.

Abstract

This paper extends the Hodge-de Rham theory of Aaron \textit{et al.} [Commun. Pure Appl. Anal. {\bf 13} (2014)] to higher-dimensional level-$l$ Sierpinski gaskets $SG_{\ell}^{n},$ providing a framework for analyzing differential forms and Laplacians on these fractal structures. We construct a sequence of graphs approximating $SG_{\ell}^{n}$ and define $k$-forms, de Rham derivatives, and their duals on these graphs. We prove that the extension of a $1$-form on a generation-$m$ graph to a $1$-form on a generation-$(m+1)$ graph is harmonic. We obtain a basis for the space of harmonic $1$-forms. We also explore the properties of $2$-forms on the level-$3$ Sierpinski gasket, under the assumptions that the $2$-forms are absolutely continuous with respect to the Kusuoka measure or the standard self-similar measure and that the Radon-Nikodym derivatives are continuous.