  We show that the upper bounds for the $L^2$-norms of $L^1$-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 $L^1$-norms of
$L^2$-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 $L^2$-norms over thin geodesic tubes.
