arXiv subjects

Anusha Bhattacharya

Publications and source records attributed to Anusha Bhattacharya.

4 recordsLinked to original sources

arXiv2026

Dirichlet Eigenvalue Approximation on Manifolds with Cylindrical Boundary

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

math.DG↗

arXiv2026

A Cheng-type Eigenvalue-Comparison Theorem for the Hodge Laplacian

We consider the class of closed Riemannian $n$-manifolds with Ricci curvature and injectivity radius bounded below by uniform constants, and an upper bound on the diameter. We establish a uniform upper bound for the eigenvalues of the Hodge Laplacian acting on differential forms on Riemannian manifolds in this class, similar to the classical eigenvalue comparison theorem proved by Cheng for the Laplace-Beltrami operator acting on smooth functions. This extends earlier work of Dodziuk and Lott, which required sectional curvature bounds in addition to bounds on other geometric quantities. As an application, we obtain uniform eigenvalue estimates for the connection Laplacian acting on $1$-forms.

math.DG↗

arXiv2026

Eigenvalue Estimates of the Hodge Laplacian Under Lower Ricci Curvature Bound

We establish uniform lower and upper bounds for the eigenvalues of the Hodge Laplacian acting on differential forms on closed Riemannian manifolds with a lower Ricci curvature bound, a positive lower bound on the injectivity radius, and an upper bound on the diameter. Our results extend earlier work of Dodziuk, Lott, and Mantuano, which required bounded sectional curvature, to the broader setting of lower Ricci curvature bounds. As applications, we obtain uniform eigenvalue bounds for the connection Laplacian acting on $1$-forms and establish a global Poincar\'e inequality for differential forms under the same geometric assumptions.

math.DG↗

arXiv2025

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

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 $(\epsilon,\rho)$-approximation of the manifold by a weighted graph, as introduced by Burago et al. By adapting their methods, we prove that as the parameters $\epsilon, \rho$ and the ratio $\frac{\epsilon}{\rho}$ approach zero, the $k$-th eigenvalue of the graph Laplacian converges uniformly to the $k$-th eigenvalue of the manifold's Laplacian for each $k$.

math.SP↗