In this talk, we present high-order approximations of the Laplace–Beltrami eigenvalue problem on point clouds. We introduce a novel geometric error analysis framework that quantifies the error arising from approximating the Riemannian metric tensor. This framework plays a foundational role in the analysis of discontinuous Galerkin (DG) methods for the Laplace–Beltrami eigenvalue problem, particularly on geometries that may exhibit discontinuities. Numerical experiments are provided to validate and illustrate the theoretical results.