Quasimode concentration on compact space forms

We show that the upper bounds for the L²-norms of L¹-normalized quasimodes that we obtained in [9] are always sharp on any compact space form. This allows us to characterize compact manifolds of constant sectional curvature using the decay rates of lower bounds of L¹-norms of L²-normalized log-quasimodes fully resolving a problem initiated by the second author and Zelditch [15]. We are also able to characterize such manifolds by the concentration of quasimodes near periodic geodesics as measured by L²-norms over thin geodesic tubes.