  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.
