Bounds on spectral gaps of Hyperbolic spin surfaces

TL;DR

This paper develops a method using spectral identities and semidefinite programming to rigorously bound the Laplacian and Dirac spectral gaps of hyperbolic spin surfaces and orbifolds, nearly reaching theoretical limits in examples.

Contribution

It introduces a novel approach combining spectral identities with semidefinite programming to bound spectral gaps of hyperbolic spin surfaces and orbifolds.

Findings

Derived nearly sharp bounds on spectral gaps for specific hyperbolic surfaces.

Identified the maximum possible Laplacian spectral gap as approximately 12.138.

Demonstrated bounds are nearly saturated by known orbifolds and surfaces.

Abstract

We describe a method for constraining Laplacian and Dirac spectra of two dimensional compact orientable hyperbolic spin manifolds and orbifolds. The key ingredient is an infinite family of identities satisfied by the spectra. These spectral identities follow from the consistency between 1) the spectral decomposition of functions on the spin bundle into irreducible representations of $\mathrm{SL}(2,\mathbb{R})$ and 2) associativity of pointwise multiplication of functions. Applying semidefinite programming methods to our identities produces rigorous upper bounds on the Laplacian spectral gap as well as on the Dirac spectral gap conditioned on the former. In several examples, our bounds are nearly sharp; a numerical algorithm based on the Selberg trace formula shows that the $[0;3,3,5]$ orbifold, a particular surface with signature $[1;3]$, and the Bolza surface nearly saturate the bounds at genus $0$, $1$ and $2$ respectively. Under additional assumptions on the number of harmonic spinors carried by the spin-surface, we obtain more restrictive bounds on the Laplacian spectral gap. In particular, these bounds apply to hyperelliptic surfaces. We also determine the set of Laplacian spectral gaps attained by all compact orientable two-dimensional hyperbolic spin orbifolds. We show that this set is upper bounded by $12.13798$; this bound is nearly saturated by the $[0;3,3,5]$ orbifold, whose first non-zero Laplacian eigenvalue is $\lambda^{(0)}_1\approx 12.13623$.