Improvements for eigenfunction averages: An application of geodesic beams

TL;DR

This paper establishes conditions under which eigenfunction averages and pointwise bounds on Riemannian manifolds can be improved logarithmically, using geodesic beam techniques to relate geometric properties to eigenfunction behavior.

Contribution

It introduces new geometric conditions on manifolds that ensure logarithmic improvements in eigenfunction averages and sup-norms, extending previous results without requiring global assumptions.

Findings

Logarithmic bounds on eigenfunction averages over submanifolds.

Logarithmic improvements for eigenfunction sup-norms under certain geometric conditions.

Application of geodesic beam techniques to derive quantitative improvements.

Abstract

Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{\phi_\lambda \}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-\Delta_g\phi_\lambda =\lambda^2 \phi_\lambda$. Given a smooth submanifold $H \subset M$ of codimension $k\geq 1$, we find conditions on the pair $(M,H)$, even when $H=\{x\}$, for which $$ \Big|\int_H\phi_\lambda d\sigma_H\Big|=O\Big(\frac{\lambda^{\frac{k-1}{2}}}{\sqrt{\log \lambda}}\Big)\qquad \text{or}\qquad |\phi_\lambda(x)|=O\Big(\frac{\lambda ^{\frac{n-1}{2}}}{\sqrt{\log \lambda}}\Big), $$ 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.