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Graph discretization of Laplacian on Riemannian manifolds with bounds on Ricci curvature

Anusha Bhattacharya, Soma Maity

Abstract

We study the approximation of eigenvalues for the Laplace-Beltrami operator on closed Riemannian manifolds in the class $\mathcal{M}$, characterized by bounded Ricci curvature, a lower bound on the injectivity radius, and an upper bound on the diameter. We use an $(ε,ρ)$-approximation of the manifold by a weighted graph, as introduced by Burago et al. By adapting their methods, we prove that as the parameters $ε, ρ$ and the ratio $\fracερ$ approach zero, the $k$-th eigenvalue of the graph Laplacian converges uniformly to the $k$-th eigenvalue of the manifold's Laplacian for each $k$.

Graph discretization of Laplacian on Riemannian manifolds with bounds on Ricci curvature

Abstract

We study the approximation of eigenvalues for the Laplace-Beltrami operator on closed Riemannian manifolds in the class , characterized by bounded Ricci curvature, a lower bound on the injectivity radius, and an upper bound on the diameter. We use an -approximation of the manifold by a weighted graph, as introduced by Burago et al. By adapting their methods, we prove that as the parameters and the ratio approach zero, the -th eigenvalue of the graph Laplacian converges uniformly to the -th eigenvalue of the manifold's Laplacian for each .

Paper Structure

This paper contains 6 sections, 26 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Some inequalities and Average dispersion
  3. An upper bound for $\lambda_{k}(\Gamma)$
  4. Smoothing Operator
  5. A lower bound for $\lambda_k(\Gamma)$
  6. Approximation of eigenfunctions

Key Result

Theorem 1.1

Consider a manifold $M\in \mathcal{M}(n,\lambda,D,i_0)$ and an $(\epsilon,\rho)$-approximation $\Gamma$ of $M$ such that $\rho<i_0$. Let $\lambda_k(M)$, $\lambda_k(\Gamma)$ denote the $k$-th eigenvalue of $-\Delta_M$ and $-\Delta_\Gamma$ respectively. Then there exists a constant $C_{\mathcal{M}}>0$

Theorems & Definitions (54)

  • Definition 1: $(\epsilon,\rho)$-approximation
  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Lemma 2.1: Laplacian comparison of the distance function
  • proof
  • Lemma 2.2
  • proof
  • Definition 2
  • Lemma 2.3
  • ...and 44 more