Quantitative Obata's theorem in discrete setting
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Acceso en línea: | |
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| _version_ | 1866914011109392384 |
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| author | Liu, Shiping Zhou, Chiyu |
| author_facet | Liu, Shiping Zhou, Chiyu |
| contents | Under mild assumptions, we show that a connected weighted graph $G$ with lower Ricci curvature bound $K>0$ in the sense of Bakry-Émery and the $d$-th non-zero Laplacian eigenvalue $λ_d$ close to $K$, with $d$ being the maximal combinatorial vertex degree of $G$, has an underlying combinatorial structure of the $d$-dimensional hypercube graph. Moreover, such a graph $G$ is close in terms of Frobenius distance to a properly weighted hypercube graph. Furthermore, we establish their closeness in terms of eigenfunctions. Our results can be viewed as discrete analogies of the almost rigidity theorem and quantitative Obata's theorem on Rimennian manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20815 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative Obata's theorem in discrete setting Liu, Shiping Zhou, Chiyu Differential Geometry Combinatorics Spectral Theory Under mild assumptions, we show that a connected weighted graph $G$ with lower Ricci curvature bound $K>0$ in the sense of Bakry-Émery and the $d$-th non-zero Laplacian eigenvalue $λ_d$ close to $K$, with $d$ being the maximal combinatorial vertex degree of $G$, has an underlying combinatorial structure of the $d$-dimensional hypercube graph. Moreover, such a graph $G$ is close in terms of Frobenius distance to a properly weighted hypercube graph. Furthermore, we establish their closeness in terms of eigenfunctions. Our results can be viewed as discrete analogies of the almost rigidity theorem and quantitative Obata's theorem on Rimennian manifolds. |
| title | Quantitative Obata's theorem in discrete setting |
| topic | Differential Geometry Combinatorics Spectral Theory |
| url | https://arxiv.org/abs/2508.20815 |