Bootstrapping Closed Hyperbolic Surfaces

TL;DR

This paper develops a bootstrap method to derive bounds on eigenvalues of the Laplace-Beltrami operator on closed hyperbolic surfaces, using spectral consistency conditions and numerical optimization.

Contribution

It introduces an efficient way to construct spectral consistency conditions and applies them to bound eigenvalues, notably providing a nearly sharp bound on the spectral gap.

Findings

Derived upper bounds on eigenvalues of hyperbolic surfaces.

Established a numerical bound on the spectral gap nearly saturated by the Bolza surface.

Demonstrated the effectiveness of bootstrap methods in spectral geometry.

Abstract

The eigenvalues of the Laplace-Beltrami operator and the integrals of products of eigenfunctions and holomorphic $s$-differentials satisfy certain consistency conditions on closed hyperbolic surfaces. These consistency conditions can be derived by using spectral decompositions to write quadruple overlap integrals in terms of triple overlap integrals in different ways. We show how to efficiently construct these consistency conditions and use them to derive upper bounds on eigenvalues, following the approach of the conformal bootstrap. As an example of such a bootstrap bound, we find a numerical upper bound on the spectral gap of closed orientable hyperbolic surfaces that is nearly saturated by the Bolza surface.