On the growth of eigenfunction averages: Microlocalization and geometry

Abstract

Let (M,g) be a smooth, compact Riemannian manifold, and let {ϕh} be an L2-normalized sequence of Laplace eigenfunctions, −h2Δgϕh=ϕh. Given a smooth submanifold H⊂M of codimension k≥1, we find conditions on the pair ({ϕh},H) for which |∫HϕhdσH|=o(h1−k2),h→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 a 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.