Spectral and Schrödinger estimates restricted to submanifolds

Abstract

Let $M$ be a complete Riemannian manifold with nonpositive sectional curvature and smooth bounded geometry. We prove logarithmic spectral projection estimates restricted to uniformly embedded submanifolds. On surfaces with pinched negative curvature, we also prove frequency-independent $L^2\to L^3$ Schrödinger restriction estimates on compact geodesic segments for times $c{\lambda }^{-1}\log \lambda$. If such a surface is even asymptotically hyperbolic, the $L^2\to L^q$ spectral estimates on complete nontrapped geodesics, for $q>2$, gain the square root of the window width. The high-frequency Schrödinger estimate holds for all times on these surfaces, both on compact segments and on complete nontrapped geodesics. Under a regularity assumption at the spectral threshold, the all-time estimate holds on the absolutely continuous spectral subspace.