Observability properties of the homogeneous wave equation on a closed manifold

We consider the wave equation on a compact Riemannian manifold ({\Omega}, g) without boundary (i.e., a closed manifold). We observe the restriction of the solutions to a measurable subset $\omega$ of {\Omega} during a time interval [0, T ] with T \textgreater{} 0. A well known result by Rauch and Taylor, and by Bardos, Lebeau and Rauch asserts that, if $\omega$ is open in {\Omega} and if the pair ($\omega$, T) satisfies the Geometric Control Condition, then an observability inequality holds comparing the total energy of solutions to the energy localized in $\omega$ x (0, T). The observability constant C T ($\omega$) is then defined as the infimum over the set of all solutions of the wave equation of the ratio of localized energy of solutions over their total energy. In this paper, we provide sharp estimates of the observability constant allowing to derive general geometric conditions guaranteeing that the wave equation is observable on $\omega$. Using the same approach, we also investigate the asymptotics of the observability constant as the observability time T tends to +$\infty$. Under topological assumptions on $\omega$, we show that 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 propagating over {\Omega}.