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.