Paper detail

Quantum ergodic restriction theorems, II: manifolds without boundary

We prove that if $(M, g)$ is a compact Riemannian manifold with ergodic geodesic flow, and if $H \subset M$ is a smooth hypersurface satisfying a generic asymmetry condition with respect to the geodesic flow, then restrictions $ϕ_j |_H$ of an orthonormal basis $\{ϕ_j\}$ of $Δ$-eigenfunctions of $(M, g)$ to $H$ are quantum ergodic on $H$. The condition on $H$ is satisfied by geodesic circles, closed horocycles and generic closed geodesics on a hyperbolic surface.