Asymptotic Expansion of the Gaussian Integral Operators on Riemannian submanifolds of $\mathbb{R}^{n}$
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909650062934016 |
|---|---|
| author | Jia-Ming Liou Lu, Chi-Chien |
| author_facet | Jia-Ming Liou Lu, Chi-Chien |
| contents | The Gaussian integral operator arises naturally as a local Euclidean approximation of the heat semigroup on a Riemannian manifold and plays a pivotal role in the analysis of graph Laplacians, particularly within the frameworks of manifold learning and spectral graph theory. In this paper, we study the asymptotic behavior of the Gaussian integral operator on a smooth Riemannian submanifold \( M \subset \mathbb{R}^n \), focusing on its expansion as \( \varepsilon \to 0^+ \). Under the assumption that the input function is real analytic near a fixed point \( x \in M \), we derive a full asymptotic expansion of the operator and compute the first-order correction term explicitly in terms of the mean curvature vector and the scalar curvature of the submanifold. In particular, we apply our results to hypersurfaces in Euclidean space and investigate geometric conditions under which points exhibit \emph{equicurvature}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_13238 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic Expansion of the Gaussian Integral Operators on Riemannian submanifolds of $\mathbb{R}^{n}$ Jia-Ming Liou Lu, Chi-Chien Differential Geometry The Gaussian integral operator arises naturally as a local Euclidean approximation of the heat semigroup on a Riemannian manifold and plays a pivotal role in the analysis of graph Laplacians, particularly within the frameworks of manifold learning and spectral graph theory. In this paper, we study the asymptotic behavior of the Gaussian integral operator on a smooth Riemannian submanifold \( M \subset \mathbb{R}^n \), focusing on its expansion as \( \varepsilon \to 0^+ \). Under the assumption that the input function is real analytic near a fixed point \( x \in M \), we derive a full asymptotic expansion of the operator and compute the first-order correction term explicitly in terms of the mean curvature vector and the scalar curvature of the submanifold. In particular, we apply our results to hypersurfaces in Euclidean space and investigate geometric conditions under which points exhibit \emph{equicurvature}. |
| title | Asymptotic Expansion of the Gaussian Integral Operators on Riemannian submanifolds of $\mathbb{R}^{n}$ |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2506.13238 |