An Approximate Inverse Spectral Theorem for Manifolds of Constant Negative Curvature

Abstract

A classical theorem of Colin de Verdi\`ere shows that on a closed manifold of fixed topology one can prescribe an arbitrary finite portion of the Laplace-Beltrami spectrum (including multiplicities, subject to the usual topological constraints) by choosing a sufficiently heterogeneous smooth metric. In this paper, we study the same inverse problem under the rigid geometric constraint of \emph{constant negative sectional curvature}. Allowing the topological complexity to vary, we prove that any finite strictly increasing target list can be approximated to arbitrary precision by a closed manifold of constant negative curvature in any dimension $d\ge2$. In $d=2$ the construction uses hyperbolic collar degeneration and the discrete spectral limit theorems of Burger, building on the collar estimates of Buser; in $d\ge3$ we build macroscopically heterogeneous hyperbolic covering manifolds assembled from ``heavy'' vertex clusters and ``long'' corridor chains whose low-energy limit is a prescribed \emph{discrete} graph Laplacian. We also record the universal obstructions at curvature normalization $\kappa\equiv -1$: Yang-Yau in $d=2$ and Kazhdan-Margulis combined with Bishop--Gromov volume comparison in $d\ge3$. In particular, $\lambda_1$ is universally bounded at $\kappa=-1$, so target lists whose first positive eigenvalue exceeds this bound cannot be approximated within the class $\kappa\equiv -1$, and accommodating arbitrarily large prescribed $\lambda_1^*$ forces $|\kappa|\to\infty$. A corollary on the arbitrarily precise prescription of scale-invariant eigenvalue ratios at $\kappa\equiv -1$ and an explicit worked example are included.