Asymptotics of the graph Laplace operator near an isolated singularity
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909982016929792 |
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| author | Pal, Susovan |
| author_facet | Pal, Susovan |
| contents | In this paper, we investigate asymptotics of the continuous graph Laplace operator on a smooth Riemannian manifold $(M,g)$ admitting an isolated singularity $x$. We show that if the curvature function $κ$ doesn't grow too fast near $x$, then the graph Laplace operator at $x$ converges to the weighted Laplace-Beltrami operator as the bandwidth $t\downarrow 0.$ On the other hand, we also prove that if one locally modifies a given Riemannian metric across $x$ by a non-constant \textit{purely angular }conformal factor, then $κ$ grows too fast and the graph Laplace operator behaves like $O(\frac{1}{\sqrt{t}})$ near $x$, as $t\downarrow 0$, given a mild condition on the angular conformal factor. We provide the Taylor expansion of the graph Laplace operator as $t\downarrow 0$ in specific cases. Numerical simulations at the end illustrate our results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13314 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotics of the graph Laplace operator near an isolated singularity Pal, Susovan Differential Geometry In this paper, we investigate asymptotics of the continuous graph Laplace operator on a smooth Riemannian manifold $(M,g)$ admitting an isolated singularity $x$. We show that if the curvature function $κ$ doesn't grow too fast near $x$, then the graph Laplace operator at $x$ converges to the weighted Laplace-Beltrami operator as the bandwidth $t\downarrow 0.$ On the other hand, we also prove that if one locally modifies a given Riemannian metric across $x$ by a non-constant \textit{purely angular }conformal factor, then $κ$ grows too fast and the graph Laplace operator behaves like $O(\frac{1}{\sqrt{t}})$ near $x$, as $t\downarrow 0$, given a mild condition on the angular conformal factor. We provide the Taylor expansion of the graph Laplace operator as $t\downarrow 0$ in specific cases. Numerical simulations at the end illustrate our results. |
| title | Asymptotics of the graph Laplace operator near an isolated singularity |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2512.13314 |