Dirichlet Eigenvalue Approximation on Manifolds with Cylindrical Boundary

TL;DR

This paper demonstrates that the Dirichlet eigenvalues of the Laplace-Beltrami operator on manifolds with cylindrical boundary can be approximated by eigenvalues of graph Laplacians constructed from proximity graphs, with convergence guarantees.

Contribution

It introduces a method to approximate Dirichlet eigenvalues on manifolds with cylindrical boundary using truncated graph Laplacians, providing uniform convergence results.

Findings

Eigenvalues of truncated graph Laplacians approximate Dirichlet eigenvalues

Convergence is uniform over a class of manifolds with bounded geometry

Approximation improves as parameters tend to zero

Abstract

We prove that the Dirichlet eigenvalues of the Laplace-Beltrami operator on a compact Riemannian manifold with cylindrical boundary can be approximated by the spectrum of truncated graph Laplacians constructed from $(\varepsilon,\rho)$-proximity graphs on the manifold. The approximation is uniform over a class $\mathcal{M}$ of manifolds, characterized by bounds on Ricci curvature, a lower bound on the injectivity radius, and an upper bound on the diameter. We show that the $k$-th eigenvalue of the truncated graph Laplacian lies between the $k$-th Dirichlet eigenvalues of truncated domains of the manifold. As the parameters $\varepsilon$ and $\rho$ and the ratio $\frac{\varepsilon}{\rho}$ tend to zero, these estimates yield convergence of the eigenvalues of the truncated graph Laplacian to the Dirichlet eigenvalues of the Laplace-Beltrami operator.