Hodge-de Rham Theory on Fractal Graphs and Fractals

TL;DR

This paper develops a new framework for differential forms on fractals, extending classical Hodge-de Rham theory to self-similar fractals like the Sierpinski gasket, and establishes connections with Kigami's Laplacian.

Contribution

It introduces a method to define k-forms and differential operators on fractals via graph approximations and limits, linking to Kigami's Laplacian and constructing explicit harmonic forms.

Findings

Laplacian on 0-forms matches Kigami's Laplacian.

Explicit harmonic 1-forms constructed for fractal examples.

Measures from 1-forms are singular with respect to Lebesgue measure.

Abstract

We present a new approach to the theory of k-forms on self-similar fractals. We work out the details for two examples, the standard Sierpinski gasket and the 3-dimensional Sierpinski gasket, but the method is expected to be effective for many PCF fractals and also infinitely ramified fractals such as the Sierpinski carpet. Our approach is to construct k-forms and de Rham differential operators d and delta for a sequence of graphs approximating the fractal and then to pass to the limit with suitable renormalization, in imitation of Kigami's approach to constructing Laplacians on functions. One of our results is that our Laplacian on 0-forms is equal to Kigami's Laplacian on functions. We give explicit construction of harmonic 1-forms for our examples. We also prove that the measures on line segments provided by 1-forms are not absolutely continuous with respect to Lebesgue measures.