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<p>We study the approximation of eigenvalues for the Laplace-Beltrami operator on closed Riemannian manifolds in the class <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {M}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">M</mi> </math></EquationSource> </InlineEquation>, characterized by bounded Ricci curvature, a lower bound on the injectivity radius, and an upper bound on the diameter. We use an <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((\epsilon ,\rho )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>ϵ</mi> <mo>,</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-approximation of the manifold by a weighted graph, as introduced by Burago et al. By adapting their methods, we prove that as the parameters <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\epsilon , \rho \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo>,</mo> <mi>ρ</mi> </mrow> </math></EquationSource> </InlineEquation> and the ratio <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\frac{\epsilon }{\rho }\)</EquationSource> <EquationSource Format="MATHML"><math> <mfrac> <mi>ϵ</mi> <mi>ρ</mi> </mfrac> </math></EquationSource> </InlineEquation> approach zero, the <i>k</i>-th eigenvalue of the graph Laplacian converges uniformly to the <i>k</i>-th eigenvalue of the manifold’s Laplacian for each <i>k</i>.</p>

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

  • 期刊论文

  • 发表日期：

    2026年2月28日

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  • Soma Maity,
  • Anusha Bhattacharya

摘要

We study the approximation of eigenvalues for the Laplace-Beltrami operator on closed Riemannian manifolds in the class \(\mathcal {M}\) M , characterized by bounded Ricci curvature, a lower bound on the injectivity radius, and an upper bound on the diameter. We use an \((\epsilon ,\rho )\) ( 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 \(\epsilon , \rho \) ϵ , ρ and the ratio \(\frac{\epsilon }{\rho }\) ϵ ρ 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.