On the growth of eigenfunction averages: microlocalization and geometry

Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{{\varphi }_h\}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-h^2{\mathrm{\Delta }}_g{\varphi }_h={\varphi }_h$. Given a smooth submanifold $H\subset M$ of codimension $k\ge 1$, we find conditions on the pair $(\{{\varphi }_h\},H)$ for which $$\mid {\int }_H{\varphi }_{h}d{\sigma }_H\mid =o(h^{\frac{1-k}{2}}),h\to 0^{+}\mathrm{.}$$ 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.