Curvature and sharp growth rates of log-quasimodes on compact manifolds

We obtain new optimal estimates for the $L^2(M)\to L^q(M)$, $q\in (2,q_c]$, $q_c=2(n+1)\mathrm{/}(n-1)$, operator norms of spectral projection operators associated with spectral windows $[\lambda ,\lambda +\delta (\lambda )]$, with $\delta (\lambda )=O((\log \lambda {)}^{-1})$ on compact Riemannian manifolds $(M,g)$ of dimension $n\ge 2$ all of whose sectional curvatures are nonpositive or negative. We show that these two different types of estimates are saturated on flat manifolds or manifolds all of whose sectional curvatures are negative. This allows us to classify compact space forms in terms of the size of $L^q$-norms of quasimodes for each Lebesgue exponent $q\in (2,q_c]$, even though it is impossible to distinguish between ones of negative or zero curvature sectional curvature for any $q>q_c$.