On the growth of eigenfunction averages: microlocalization and geometry

TL;DR

This paper investigates how eigenfunctions of the Laplacian on compact Riemannian manifolds behave when integrated over submanifolds, establishing conditions under which these averages decay faster than a certain rate, especially in negatively curved geometries.

Contribution

It provides new conditions on eigenfunctions and submanifolds that guarantee decay of averages, linking microlocal properties and geometry to eigenfunction behavior.

Findings

Decay of eigenfunction averages on submanifolds under certain geometric conditions

Recurrent conormal directions have measure zero for decay to hold

Results apply to surfaces with Anosov flow and constant negative curvature

Abstract

Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{\phi_h\}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-h^2\Delta_g\phi_h=\phi_h$. Given a smooth submanifold $H \subset M$ of codimension $k\geq 1$, we find conditions on the pair $(\{\phi_h\},H)$ for which $$ \Big|\int_H\phi_hd\sigma_H\Big|=o(h^{\frac{1-k}{2}}),\qquad h\to 0^+. $$ One such condition is that the set of conormal directions to $H$ that are recurrent has measure $0$. In particular, we show that the upper bound holds for any $H$ if $(M,g)$ is surface with Anosov geodesic flow or a manifold of constant negative curvature. The results are obtained by characterizing the behavior of the defect measures of eigenfunctions with maximal averages.