PreprintCC0arXiv
A Weighted Discretization of Riemannian Manifolds with Lower Ricci Bounds
Abstract
Let (M,g) be a connected, compact, n-dimensional Riemannian manifold with Ric(M,g)≥−(n−1)κg. We introduce a weighted combinatorial Laplacian on ε-discretizations of M and prove a spectral comparison theorem between the weighted graph Laplacian and the Laplace-Beltrami operator. More precisely, the eigenvalues of the two operators are uniformly comparable with constants depending only on n,κ,ε, independently of the injectivity radius. As an application, we prove spectral stability under measured Gromov-Hausdorff convergence. We also recover the Schoen-Wolpert-Yau inequality using the weighted discretization on families of pinching genus-$2$ hyperbolic surfaces.
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BibTeX
@misc{tiwari2026weighted,
title = {{A Weighted Discretization of Riemannian Manifolds with Lower Ricci Bounds}},
author = {Aditya Tiwari},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.02405},
eprint = {2608.02405},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.02405},
note = {CC0},
}
RIS
TY - UNPB
TI - A Weighted Discretization of Riemannian Manifolds with Lower Ricci Bounds
AU - Aditya Tiwari
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2608.02405
UR - https://arxiv.org/abs/2608.02405
AB - Let (M,g) be a connected, compact, n-dimensional Riemannian manifold with Ric(M,g)≥−(n−1)κg. We introduce a weighted combinatorial Laplacian on ε-discretizations of M and prove a spectral comparison theorem between the weighted graph Laplacian and the Laplace-Beltrami operator. More precisely, the eigenvalues of the two operators are uniformly comparable with constants depending only on n,κ,ε, independently of the injectivity radius. As an application, we prove spectral stability under measured Gromov-Hausdorff convergence. We also recover the Schoen-Wolpert-Yau inequality using the weighted discretization on families of pinching genus-$2$ hyperbolic surfaces.
ER -