Graph discretization of Laplacian on Riemannian manifolds with bounds on Ricci curvature
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908859405172736 |
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| author | Bhattacharya, Anusha Maity, Soma |
| author_facet | Bhattacharya, Anusha Maity, Soma |
| contents | We study the approximation of eigenvalues for the Laplace-Beltrami operator on closed Riemannian manifolds in the class $\mathcal{M}$, characterized by bounded Ricci curvature, a lower bound on the injectivity radius, and an upper bound on the diameter. We use an $(ε,ρ)$-approximation of the manifold by a weighted graph, as introduced by Burago et al. By adapting their methods, we prove that as the parameters $ε, ρ$ and the ratio $\fracερ$ approach zero, the $k$-th eigenvalue of the graph Laplacian converges uniformly to the $k$-th eigenvalue of the manifold's Laplacian for each $k$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_18323 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Graph discretization of Laplacian on Riemannian manifolds with bounds on Ricci curvature Bhattacharya, Anusha Maity, Soma Spectral Theory Differential Geometry 58J50, 58J60, 65N25, 53C21, 05C50 We study the approximation of eigenvalues for the Laplace-Beltrami operator on closed Riemannian manifolds in the class $\mathcal{M}$, characterized by bounded Ricci curvature, a lower bound on the injectivity radius, and an upper bound on the diameter. We use an $(ε,ρ)$-approximation of the manifold by a weighted graph, as introduced by Burago et al. By adapting their methods, we prove that as the parameters $ε, ρ$ and the ratio $\fracερ$ approach zero, the $k$-th eigenvalue of the graph Laplacian converges uniformly to the $k$-th eigenvalue of the manifold's Laplacian for each $k$. |
| title | Graph discretization of Laplacian on Riemannian manifolds with bounds on Ricci curvature |
| topic | Spectral Theory Differential Geometry 58J50, 58J60, 65N25, 53C21, 05C50 |
| url | https://arxiv.org/abs/2501.18323 |