Strichartz estimates and the nonlinear Schr\"odinger equation on manifolds with boundary

We establish Strichartz estimates for the Schr"odinger equation on Riemannian manifolds $(\mathrm{\Omega },\text{g})$ with boundary, for both the compact case and the case that $\mathrm{\Omega }$ is the exterior of a smooth, non-trapping obstacle in Euclidean space. The estimates for exterior domains are scale invariant; the range of Lebesgue exponents $(p,q)$ for which we obtain these estimates is smaller than the range known for Euclidean space, but includes the key $L_t^4{L}_x^{\mathrm{\infty }}$ estimate, which we use to give a simple proof of well-posedness results for the energy critical Schr"odinger equation in 3 dimensions. Our estimates on compact manifolds involve a loss of derivatives with respect to the scale invariant index. We use these to establish well-posedness for finite energy data of certain semilinear Schr"odinger equations on general compact manifolds with boundary.