Linear programming bounds for hyperbolic surfaces

TL;DR

This paper adapts linear programming techniques from sphere packings to hyperbolic surfaces, establishing new upper bounds on geometric and spectral properties, and providing lower bounds on systoles for certain genera.

Contribution

It introduces a novel application of linear programming to hyperbolic geometry, improving bounds on systole, eigenvalues, and related invariants of hyperbolic surfaces.

Findings

New upper bounds on systole and eigenvalues for hyperbolic surfaces.

Bounds are the best known for low genus and as genus increases.

Lower bounds on systole imply spectral gaps larger than 1/4 for certain genera.

Abstract

We adapt linear programming methods from sphere packings to closed hyperbolic surfaces and obtain new upper bounds on their systole, their kissing number, the first positive eigenvalue of their Laplacian, the multiplicity of their first eigenvalue, and their number of small eigenvalues. Apart from a few exceptions, the resulting bounds are the current best known both in low genus and as the genus tends to infinity. Our methods also provide lower bounds on the systole (achieved in genus $2$ to $7$, $14$, and $17$) that are sufficient for surfaces to have a spectral gap larger than $1/4$.