Global in time Strichartz inequalities on asymptotically flat manifolds   with temperate trapping
  We prove global Strichartz inequalities for the Schr\"odinger equation on a large class of asymptotically conical manifolds. Letting $ P $ be the nonnegative Laplace operator and $ f_0 \in C_0^{\infty}({\mathbb R}) $ be a smooth cutoff equal to $1$ near zero, we show first that the low frequency part of any solution $ e^{-itP} u_0 $, i.e. $ f_0 (P) e^{-itP} u_0 $, enjoys the same global Strichartz estimates as on $ {\mathbb R}^n $ in dimension $ n \geq 3 $. We also show that the high energy part $ (1-f_0)(P) e^{-itP} u_0$ also satisfies global Strichartz estimates without loss of derivatives outside a compact set, even if the manifold has trapped geodesics but in a temperate sense. We then show that the full solution $ e^{-itP}u_0 $ satisfies global space-time Strichartz estimates if the trapped set is empty or sufficiently filamentary, and we derive a scattering theory for the $ L^2 $ critical nonlinear Schr\"odinger equation in this geometric framework. 
Jean-Marc Bouclet and Haruya Mizutani
math.AP