Asymptotics of the graph Laplace operator near an isolated singularity

2025/12/15 by Pal, Susovan
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2512.13314

Abstract

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 purely angular conformal factor, then κ grows too fast and the graph Laplace operator behaves like O((1)/(√(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.

Citations

• Manifolds with kinks and the asymptotic behavior of the graph Laplacian operator with Gaussian kernel
• Boundary Estimation from Point Clouds: Algorithms, Guarantees and\n Applications
• Ghost Point Diffusion Maps for solving elliptic PDE's on Manifolds with Classical Boundary Conditions
• Diffusion Maps for Embedded Manifolds with Boundary with Applications to PDEs
• Density Estimation on Manifolds with Boundary
• Graph Laplacians and their convergence on random neighborhood graphs
• Towards a theoretical foundation for Laplacian-based manifold methods

Related

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• Graph approximations to the Laplacian spectra
• First order operators on shrinking graph-like spaces
• An Analysis of the Convergence of Graph Laplacians
• Spectral convergence of graph Laplacians with Ricci curvature bounds and in non-collapsed Ricci limit spaces