Observability properties of the homogeneous wave equation on a closed manifold

We consider the wave equation on a closed Riemannian manifold. We observe the restriction of the solutions to a measurable subset $\omega$ along a time interval $[0, T]$ with $T>0$. It is well known that, if $\omega$ is open and if the pair $(\omega,T)$ satisfies the Geometric Control Condition then an observability inequality is satisfied, comparing the total energy of solutions to their energy localized in $\omega \times (0, T)$. The observability constant $C\_T({\omega})$ is then defined as the infimum over the set of all nontrivial solutions of the wave equation of the ratio of localized energy of solutions over their total energy. In this paper, we provide estimates of the observability constant based on a low/high frequency splitting procedure allowing us to derive general geometric conditions guaranteeing that the wave equation is observable on a measurable subset $\omega$. We also establish that, as $T\rightarrow+\infty$, the ratio $C\_T({\omega})/T$ converges to the minimum of two quantities: the first one is of a spectral nature and involves the Laplacian eigenfunctions, the second one is of a geometric nature and involves the average time spent in $\omega$ by Riemannian geodesics.