Gespeichert in:
| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2508.12319 |
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Inhaltsangabe:
- 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.