Spectral Bounds on Hyperbolic 3-Manifolds: Associativity and the Trace Formula

TL;DR

This paper develops rigorous spectral bounds for hyperbolic 3-manifolds using linear programming, spectral identities, and the trace formula, providing new constraints on eigenvalues, spectral gaps, and geometric features.

Contribution

It introduces a novel combination of bootstrap-inspired spectral identities and the trace formula to derive explicit bounds on eigenvalues and geometric parameters of hyperbolic 3-manifolds.

Findings

Upper bounds on the first and second Laplacian eigenvalues.

An explicit upper bound of 47.32 on the spectral gap for all closed hyperbolic 3-manifolds.

Bounds are nearly sharp when compared with numerical estimates of example orbifolds.

Abstract

We constrain the low-energy spectra of Laplace operators on closed hyperbolic manifolds and orbifolds in three dimensions, including the standard Laplace-Beltrami operator on functions and the Laplacian on powers of the cotangent bundle. Our approach employs linear programming techniques to derive rigorous bounds by leveraging two types of spectral identities. The first type, inspired by the conformal bootstrap, arises from the consistency of the spectral decomposition of the product of Laplace eigensections, and involves the Laplacian spectra as well as integrals of triple products of eigensections. We formulate these conditions in the language of representation theory of $\mathrm{PSL}_2(\mathbb{C})$ and use them to prove upper bounds on the first and second Laplacian eigenvalues. The second type of spectral identities follows from the Selberg trace formula. We use them to find upper bounds on the spectral gap of the Laplace-Beltrami operator on hyperbolic 3-orbifolds, as well as on the systole length of hyperbolic 3-manifolds, as a function of the volume. Further, we prove that the spectral gap $\lambda_1$ of the Laplace-Beltrami operator on all closed hyperbolic 3-manifolds satisfies $\lambda_1 < 47.32$. Along the way, we use the trace formula to estimate the low-energy spectra of a large set of example orbifolds and compare them with our general bounds, finding that the bounds are nearly sharp in several cases.