Linear programming bounds for hyperbolic surfaces

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 $1\mathrm{/}4$.