Growth of high Lp norms for eigenfunctions : an application of geodesic beams

This work concerns $L^p$ norms of high energy Laplace eigenfunctions: $(-{\mathrm{\Delta }}_g-{\lambda }^2){\varphi }_{\lambda }=0$, $\parallel {\varphi }_{\lambda }{\parallel }_{L^2}=1$. Sogge (1988) gave optimal estimates on the growth of $\parallel {\varphi }_{\lambda }{\parallel }_{L^p}$ for a general compact Riemannian manifold. Here we give general dynamical conditions guaranteeing quantitative improvements in $L^p$ estimates for $p>p_c$, where $p_c$ is the critical exponent. We also apply results of an earlier paper (Canzani and Galkowski 2018) to obtain quantitative improvements in concrete geometric settings including all product manifolds. These are the first results giving quantitative improvements for estimates on the $L^p$ growth of eigenfunctions that only require dynamical assumptions. In contrast with previous improvements, our assumptions are local in the sense that they depend only on the geodesics passing through a shrinking neighborhood of a given set in $M$. Moreover, we give a structure theorem for eigenfunctions which saturate the quantitatively improved $L^p$ bound. Modulo an error, the theorem describes these eigenfunctions as finite sums of quasimodes which, roughly, approximate zonal harmonics on the sphere scaled by $1∕\sqrt{\log <!--FUNCTION\; APPLICATION-->\lambda }$.