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

2025/08/17 by Ngai, Sze-Man, Zhou, Shui-Hong
#28A80 #58A10 #58A12 #58A14 #Differential Geometry (math.DG) #FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2508.12319

Abstract

This paper extends the Hodge-de Rham theory of Aaron et al. [Commun. Pure Appl. Anal. \bf 13 (2014)] to higher-dimensional level-l Sierpinski gaskets SG_ℓⁿ, providing a framework for analyzing differential forms and Laplacians on these fractal structures. We construct a sequence of graphs approximating SG_ℓⁿ 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.

Citations

• The Kusuoka measure and the energy Laplacian on level-k Sierpiński gaskets
• Hodge-de Rham Theory on Fractal Graphs and Fractals

Related

• Hodge-de Rham Theory on Fractal Graphs and Fractals
• Measurable Riemannian structure on higher dimensional harmonic Sierpinski gaskets
• A note on energy forms on fractal domains
• Harmonic embeddings of the stretched Siepinski gasket
• The Kusuoka measure and the energy Laplacian on level-k Sierpiński gaskets