Improvements for eigenfunction averages: An application of geodesic beams

Abstract

Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{{\varphi }_{\lambda }\}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-{\Delta }_g{\varphi }_{\lambda }={\lambda }^2{\varphi }_{\lambda }$. Given a smooth submanifold $H\subset M$ of codimension $k\ge 1$, we find conditions on the pair $(M,H)$, even when $H=\{x\}$, for which $$\left|{\int }_H{\varphi }_{\lambda }d{\sigma }_H\right|=O\left(\frac{{\lambda }^{\frac{k-1}{2}}}{\sqrt{\log \lambda }}\right)\text{or}|{\varphi }_{\lambda }(x)|=O\left(\frac{{\lambda }^{\frac{n-1}{2}}}{\sqrt{\log \lambda }}\right),$$ as $\lambda \to \infty$. These conditions require no global assumption on the manifold $M$ and instead relate to the structure of the set of recurrent directions in the unit normal bundle to $H$. Our results extend all previously known conditions guaranteeing improvements on averages, including those on sup-norms. For example, we show that if $(M,g)$ is a surface with Anosov geodesic flow, then there are logarithmically improved averages for any $H\subset M$. We also find weaker conditions than having no conjugate points which guarantee $\sqrt{\log \lambda }$ improvements for the $L^{\infty }$ norm of eigenfunctions. Our results are obtained using geodesic beam techniques, which yield a mechanism for obtaining general quantitative improvements for averages and sup-norms.

arXiv comment70 pages, 4 figures. The new version includes a small revision to the arguments in section 6

Cited by in corpus (1)

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