Graph discretization of Laplacian on Riemannian manifolds with bounded Ricci curvature

Abstract: We study the approximation of eigenvalues and eigenfunctions for the Laplace-Beltrami operator on compact manifolds without boundary. The analysis is centered on manifolds characterized by specific geometric constraints: lower bounds on Ricci curvature and injectivity radius, and an upper bound on diameter. Using weighted graph techniques, we approximate the eigenvalues for manifolds having a uniform lower bound on the volume of small balls and also investigate uniform bounds of eigenvalues applicable across the entire class.