Laplace–Beltrami eigenvalue problems arise naturally in a wide range of applications, including shape analysis, surface vibration, geometric data processing, computer graphics, and manifold learning. We develop high-order numerical approximations for these problems on point clouds. A novel geometric error analysis framework is introduced to quantify the errors caused by approximating the Riemannian metric tensor. This framework provides a rigorous foundation for the analysis of discontinuous Galerkin methods on discrete geometries, including geometries with discontinuous approximations. Numerical experiments verify the theoretical results and demonstrate the accuracy and effectiveness of the proposed methods.