Let be a smooth, compact Riemannian manifold, and let be an -normalized sequence of Laplace eigenfunctions, . Given a smooth submanifold of codimension , we find conditions on the pair for which One such condition is that the set of conormal directions to that are recurrent has measure . In particular, we show that the upper bound holds for any if 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.
"On the growth of eigenfunction averages: Microlocalization and geometry." Duke Math. J. 168 (16) 2991 - 3055, 1 November 2019. https://doi.org/10.1215/00127094-2019-0020