Averages of Eigenfunctions Over Hypersurfaces

Let (M, g) be a compact, smooth, Riemannian manifold and \({\{ \phi_h \}}\) an L2-normalized sequence of Laplace eigenfunctions with defect measure \({\mu}\). Let H be a smooth hypersurface with unit exterior normal \(\nu\). Our main result says that when \(\mu\) is not concentrated conormally to H, the eigenfunction restrictions to H satisfy $$\int_H \phi_h d\sigma_H = o(1) \quad {\rm and} \quad \int_H h D_{\nu} \phi_h d\sigma_H = o(1),$$ \({h \to 0^+}\).