Bootstrap Bounds on Closed Hyperbolic Manifolds

TL;DR

This paper develops bootstrap bounds on eigenvalues and eigenfunction integrals of closed hyperbolic manifolds, using spectral consistency conditions and semidefinite programming to derive new geometric inequalities.

Contribution

It introduces a novel bootstrap approach to bound spectral data of hyperbolic manifolds, analogous to conformal bootstrap in quantum field theory.

Findings

Upper bound on the gap between consecutive eigenvalues

Bounds on integrals of eigenfunctions and eigentensors

Analytic eigenvalue inequality for hyperbolic surfaces

Abstract

The eigenvalues of the Laplace-Beltrami operator and the integrals of products of eigenfunctions must satisfy certain consistency conditions on compact Riemannian manifolds. These consistency conditions are derived by using spectral decompositions to write quadruple overlap integrals in terms of products of triple overlap integrals in multiple ways. In this paper, we show how these consistency conditions imply bounds on the Laplacian eigenvalues and triple overlap integrals of closed hyperbolic manifolds, in analogy to the conformal bootstrap bounds on conformal field theories. We find an upper bound on the gap between two consecutive nonzero eigenvalues of the Laplace-Beltrami operator in terms of the smaller eigenvalue, an upper bound on the smallest eigenvalue of the rough Laplacian on symmetric, transverse-traceless, rank-2 tensors, and bounds on integrals of products of eigenfunctions and eigentensors. Our strongest bounds involve numerically solving semidefinite programs and are presented as exclusion plots. We also prove the analytic bound $\lambda_{i+1} \leq 1/2+3 \lambda_i+\sqrt{\lambda_i^2+2 \lambda_i+1/4}$ for consecutive nonzero eigenvalues of the Laplace-Beltrami operator on closed orientable hyperbolic surfaces. We give examples of genus-2 surfaces that nearly saturate some of these bounds. To derive the consistency conditions, we make use of a transverse-traceless decomposition for symmetric tensors of arbitrary rank.