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 $\mathcal{M}$, characterized by bounded Ricci curvature, a lower bound on the injectivity radius, and an upper bound on the diameter. We use an $(\epsilon,\rho)$-approximation of the manifold by a weighted graph, as introduced by Burago et al. By adapting their methods, we prove that as the parameters $\epsilon, \rho$ and the ratio $\frac{\epsilon}{\rho}$ approach zero, the $k$-th eigenvalue of the graph Laplacian converges uniformly to the $k$-th eigenvalue of the manifold's Laplacian for each $k$.