Eigenfunctions of the Laplacian cannot vanish on a set of positive measure. Quantitative versions of this unique continuation are well-known on fixed Riemannian manifolds: the L² norm of an eigenfunction is bounded by its L² norm on a set of positive measure times a constant which grows exponentially with the frequency. This growing rate is sharp and reflects in observability properties for the heat equation.

In this talk, I will present recent results, in collaboration with M. Rouveyrol (Orsay) about non-compact hyperbolic surfaces. Quantitative unique continuation, and observability of the heat equation, hold under a necessary and sufficient condition of thickness of the observed set: it must intersect every large enough metric ball with a mass bounded from below, proportionally to the mass of the ball itself. The proof crucially uses the Logunov-Mallinikova estimates.