Exterior mass estimates and $L^2$ restriction bounds for Neumann data along hypersurfaces

We study the problem of estimating the $L^2$ norm of Laplace eigenfunctions on a compact Riemannian manifold $M$ when restricted to a hypersurface $H$. We prove mass estimates for the restrictions of eigenfunctions $phi_h$, $(h^2 Delta - 1)phi_h = 0$, to $H$ in the region exterior to the coball bundle of $H$, on $h^{delta}$-scales ($0leq delta < 2/3$). We use this estimate to obtain an $O(1)$ $L^2$-restriction bound for the Neumann data along $H.$ The estimate also applies to eigenfunctions of semiclassical Schrodinger operators.