Graph discretization of Laplacian on Riemannian manifolds with bounds on Ricci curvature

2025/01/30 by Bhattacharya, Anusha, Maity, Soma
#Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2501.18323

Abstract

We study the approximation of eigenvalues for the Laplace-Beltrami operator on closed Riemannian manifolds in the class \MMM, 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.

Related

• A graph discretization of the Laplace-Beltrami operator
• Spectral convergence of graph Laplacians with Ricci curvature bounds and in non-collapsed Ricci limit spaces
• Graph approximations to the Laplacian spectra
• Eigenvalue estimates for Beltrami-Laplacian under Bakry-Émery Ricci curvature condition
• Approximation of the spectrum of a manifold by discretization