What does this research mean for the field?

The Dirichlet eigenvalues of the Laplace–Beltrami operator on compact Riemannian manifolds with cylindrical boundary can be uniformly approximated by the spectrum of truncated graph Laplacians derived from proximity graphs. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

The aim is to show how Dirichlet eigenvalues can be approximated using graph Laplacians on specific manifolds.

March 17, 2026Open Access

Dirichlet Eigenvalue Approximation on Manifolds with Cylindrical Boundary

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The aim is to show how Dirichlet eigenvalues can be approximated using graph Laplacians on specific manifolds.
Proved the approximation of Dirichlet eigenvalues via truncated graph Laplacians.
Constructed proximity graphs based on parameters ε and ρ.
Characterized manifolds using bounds on Ricci curvature, injectivity radius, and diameter.
Showed that the $k$-th eigenvalue of the graph Laplacian is bounded by Dirichlet eigenvalues.
Established uniform approximation across a class of manifolds.
Demonstrated convergence of eigenvalues as ε, ρ approach zero.

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Implication

We prove that graph Laplacians approximate eigenvalues in Riemannian manifolds, suggesting broader applications.

Cite This Study

Atanu Bhattacharya (2026) studied this question.

synapsesocial.com/papers/69b8f12fdeb47d591b8c628e https://doi.org/10.5281/zenodo.19027018

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Dirichlet Eigenvalue Approximation on Manifolds with Cylindrical Boundary

Key Points

Discussion

Implication

Cite This Study

Dirichlet Eigenvalue Approximation on Manifolds with Cylindrical Boundary

Key Points

Discussion

Implication

Cite This Study

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