The first nontrivial eigenvalue of the Laplacian on a closed Riemannian manifold (called the spectral gap) is a geometrical invariant that encodes many interesting properties of the manifold. Notably, it gives some quantitative information on the connectedness of the manifold, and on the rate of convergence to equilibrium of the heat flow.

 

I will focus on hyperbolic surfaces, which are closed manifolds of dimension 2 with constant curvature -1. After defining these objects, I will review some results about the spectral gap in this setting. Recent developments provide probabilistic constructions of random surfaces that display optimal spectral gaps.