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

Abstract

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