Averages of eigenfunctions over hypersurfaces

Let $(M,g)$ be a compact, smooth, Riemannian manifold and $\{{\varphi }_h\}$ an $L^2$-normalized sequence of Laplace eigenfunctions with defect measure $\mu$. Let $H$ be a smooth hypersurface. Our main result says that when $\mu$ is $\mathit{\text{not}}$ concentrated conormally to $H$, the eigenfunction restrictions to $H$ and the restrictions of their normal derivatives to $H$ have integrals converging to 0 as $h\to 0^{+}$.