Averages of eigenfunctions over hypersurfaces (1705.09595v2)

Abstract: Let $(M,g)$ be a compact, smooth, Riemannian manifold and ${ \phi_h }$ 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 $\textit{not}$ concentrated conormally to $H$, the eigenfunction restrictions to $H$ and the restrictions of their normal derivatives to $H$ have integrals converging to 0 as $h \to 0^+$.