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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2508.12319 |
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| _version_ | 1866913995510775808 |
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| author | Ngai, Sze-Man Zhou, Shui-Hong |
| author_facet | Ngai, Sze-Man Zhou, Shui-Hong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hodge-de Rham Theory on Higher-Dimensional Level-L Sierpinski Gaskets Ngai, Sze-Man Zhou, Shui-Hong Differential Geometry K-Theory and Homology 28A80, 58A14, 58A12, 58A10 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. |
| title | Hodge-de Rham Theory on Higher-Dimensional Level-L Sierpinski Gaskets |
| topic | Differential Geometry K-Theory and Homology 28A80, 58A14, 58A12, 58A10 |
| url | https://arxiv.org/abs/2508.12319 |