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An Approximate Inverse Spectral Theorem for Manifolds of Constant Negative Curvature

Mayukh Mukherjee

Abstract

A classical theorem of Colin de Verdière 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 $κ\equiv -1$: Yang-Yau in $d=2$ and Kazhdan-Margulis combined with Bishop--Gromov volume comparison in $d\ge3$. In particular, $λ_1$ is universally bounded at $κ=-1$, so target lists whose first positive eigenvalue exceeds this bound cannot be approximated within the class $κ\equiv -1$, and accommodating arbitrarily large prescribed $λ_1^*$ forces $|κ|\to\infty$. A corollary on the arbitrarily precise prescription of scale-invariant eigenvalue ratios at $κ\equiv -1$ and an explicit worked example are included.

An Approximate Inverse Spectral Theorem for Manifolds of Constant Negative Curvature

Abstract

A classical theorem of Colin de Verdière 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 . In the construction uses hyperbolic collar degeneration and the discrete spectral limit theorems of Burger, building on the collar estimates of Buser; in 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 : Yang-Yau in and Kazhdan-Margulis combined with Bishop--Gromov volume comparison in . In particular, is universally bounded at , so target lists whose first positive eigenvalue exceeds this bound cannot be approximated within the class , and accommodating arbitrarily large prescribed forces . A corollary on the arbitrarily precise prescription of scale-invariant eigenvalue ratios at and an explicit worked example are included.
Paper Structure (17 sections, 16 theorems, 87 equations)

This paper contains 17 sections, 16 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Spectral Universality of Discrete Complete Graphs
  3. The Surface Case (d=2): Degenerating Collars
  4. Higher Dimensions (d>=3): Heavy-Vertex Coverings and Discrete Limits
  5. A base manifold surjecting onto a free group
  6. Cell problem and effective conductances
  7. Macroscopic networks with heavy vertices
  8. Expander wiring inside clusters (with exposed ports).
  9. The global Schreier covering.
  10. Discrete scaling (graph model)
  11. Spectral reduction: from Mm to Lm
  12. Proof of the Main Theorem
  13. Geometric Obstructions to Bounded Curvature
  14. Surfaces (d=2)
  15. Higher dimensions (d>=3)
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $d \ge 2$ be an integer, and let $0 = \lambda^*_0 < \lambda^*_1 < \dotsb < \lambda^*_n$ be a finite, strictly increasing sequence of real numbers. For any error tolerance $\varepsilon > 0$, there exists a closed, connected $d$-dimensional Riemannian manifold $(M, g)$ such that: Moreover, there exists a constant $\Lambda_d>0$ (see Section sec:obstruction) such that every closed $d$-manifold of

Theorems & Definitions (45)

  • Theorem 1.1: Approximate Inverse Spectral Theorem
  • Remark 1.2: Topology of the approximating manifolds
  • Remark 1.3: Geometric collapse
  • Corollary 1.4: Corollary \ref{['cor:eigenvalue_ratios']}, stated informally
  • Remark 1.5: Simple spectra only
  • Remark 1.6: Countability obstruction to exact prescription in $d\ge 3$
  • Lemma 2.1: Colin de Verdière CdV1988
  • proof
  • Remark 2.2: Scaling with constant vertex measure
  • Theorem 3.1: Burger Burger1988Burger1990; Buser Buser1992
  • ...and 35 more