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

Abstract

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 ${\varphi }_h$, $(h^2\Delta -1){\varphi }_h=0$, to $H$ in the region exterior to the coball bundle of $H$, on $h^{\delta }$-scales ($0\le \delta <2/3$). 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 Schrödinger operators.

22 pages. Second version has (sharp) improved restriction estimate following contributions from A. Hassell (added as an author). Version 3 fixes some mistakes and typos, and adds a more general result on semiclassical Schrodinger operator eigenfunctions