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Dirichlet Eigenvalue Approximation on Manifolds with Cylindrical Boundary

Anusha Bhattacharya

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 $(\varepsilon,ρ)$-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 $ρ$ and the ratio $\frac{\varepsilon}ρ$ tend to zero, these estimates yield convergence of the eigenvalues of the truncated graph Laplacian to the Dirichlet eigenvalues of the Laplace-Beltrami operator.

Dirichlet Eigenvalue Approximation on Manifolds with Cylindrical Boundary

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 -proximity graphs on the manifold. The approximation is uniform over a class 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 -th eigenvalue of the truncated graph Laplacian lies between the -th Dirichlet eigenvalues of truncated domains of the manifold. As the parameters and and the ratio tend to zero, these estimates yield convergence of the eigenvalues of the truncated graph Laplacian to the Dirichlet eigenvalues of the Laplace-Beltrami operator.
Paper Structure (8 sections, 18 theorems, 74 equations)

This paper contains 8 sections, 18 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Discretization Scheme
  3. Preliminaries
  4. Min-Max Characterization
  5. Average Dispersion
  6. Discretization of the manifold
  7. Smoothing operator
  8. Interpolation map

Key Result

Theorem 1.2

For $0 < \varepsilon < \rho < t < s_0$, consider a manifold $M\in \mathcal{M}$ . Let $\lambda_k(M_{t+\varepsilon})$ and $\lambda_k(M_{t+2\varepsilon-\rho})$ denote the Dirichlet eigenvalues of the Laplace-Beltrami operator restricted to the truncated domains $M_{t+\varepsilon}$ and $M_{t+2\varepsilo

Theorems & Definitions (44)

  • Definition 1
  • Definition 2: $(\varepsilon,\rho)$-proximity graph
  • Remark 1.1
  • Definition 3
  • Theorem 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Definition 4
  • Lemma 2.1
  • proof
  • ...and 34 more