Asymptotic Expansion of the Gaussian Integral Operators on Riemannian submanifolds of ℝⁿ

2025/06/16 by Jia-Ming, Liou, Chi-Chien Lu +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory in Mathematical Physics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2506.13238

openalex publication_date 2025/06/16 · openalex created_date 2025/10/13 · openalex updated_date 2026/07/28

Abstract

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 \( M ⊂ ℝⁿ \), focusing on its expansion as \( ε → 0⁺ \). Under the assumption that the input function is real analytic near a fixed point \( x ∈ M \), 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 equicurvature.

Citations

• Towards a theoretical foundation for Laplacian-based manifold methods
• Laplacian Eigenmaps for Dimensionality Reduction and Data Representation

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