An Approximate Inverse Spectral Theorem for Manifolds of Constant Negative Curvature

TL;DR

This paper demonstrates that any finite spectral list can be approximated by hyperbolic manifolds of constant negative curvature, revealing new inverse spectral results under geometric constraints across various dimensions.

Contribution

It extends inverse spectral theory to constant negative curvature manifolds, constructing manifolds with prescribed spectral properties using hyperbolic degeneration and covering techniques.

Findings

Any finite spectral list can be approximated by hyperbolic manifolds.

Universal obstructions like Yang-Yau and Kazhdan-Margulis are identified.

The first eigenvalue is bounded at curvature -1, affecting spectral approximation.

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.