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

TL;DR

This paper demonstrates that weighted graph approximations of Riemannian manifolds with bounded Ricci curvature can accurately converge to the manifold's Laplacian eigenvalues, providing a bridge between discrete and continuous spectral geometry.

Contribution

It extends graph discretization techniques to manifolds with Ricci curvature bounds, proving eigenvalue convergence under specific approximation parameters.

Findings

Eigenvalues of graph Laplacian converge to manifold Laplacian eigenvalues

Convergence is uniform for each fixed eigenvalue index k

Results apply to manifolds with bounded Ricci curvature, injectivity radius, and diameter

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 $k$.