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This work concerns $L^p$ norms of high energy Laplace eigenfunctions,$(-\Delta_g-\lambda^2)\phi_\lambda=0$, $\|\phi_\lambda\|_{L^2}=1$. In 1988,Sogge gave optimal estimates on the growth of $\|\phi_\lambda\|_{L^p}$ for ageneral compact Riemannian manifold. The goal of this article is to givegeneral dynamical conditions guaranteeing quantitative improvements in $L^p$estimates for $p>p_c$, where $p_c$ is the critical exponent. We also applyprevious results of the authors to obtain quantitative improvements in concretegeometric settings including all product manifolds. These are the first resultsimproving estimates for the $L^p$ growth of eigenfunctions that only requiredynamical assumptions. In contrast with previous improvements, our assumptionsare local in the sense that they depend only on the geodesics passing through ashrinking neighborhood of a given set in $M$. Moreover, the article gives astructure theorem for eigenfunctions which saturate the quantitatively improved$L^p$ bound. Modulo an error, the theorem describes these eigenfunctions asfinite sums of quasimodes which, roughly, approximate zonal harmonics on thesphere scaled by $1/\sqrt{\log \lambda}$.

Growth of high $L^p$ norms for eigenfunctions: an application of geodesic beams