ON THE GROWTH OF EIGENFUNCTION AVERAGES: MICROLOCALIZATION AND GEOMETRY

Let (M, g) be a smooth, compact Riemannian manifold, and let {phi(h)} be an L-2- normalized sequence of Laplace eigenfunctions, -h(2) Delta g phi(h) = phi(h). Given a smooth submanifold H subset of M of codimension k >= 1, we find conditions on the pair ({phi(h)}, H) for which vertical bar integral(H)phi(h) d sigma(H)vertical bar = o(h (1-k/2)), 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.