Strichartz estimates for the Schrödinger equation on negatively curved compact manifolds

Abstract: We obtain improved Strichartz estimates for solutions of the Schr\"odinger equation on negatively curved compact manifolds which improve the classical universal results results of Burq, G\'erard and Tzvetkov [11] in this geometry. In the case where the spatial manifold is a hyperbolic surface we are able to obtain no-loss $L^{q_c}_{t,x}$-estimates on intervals of length $\log \lambda\cdot \lambda^{-1} $ for initial data whose frequencies are comparable to $\lambda$, which, given the role of the Ehrenfest time, is the natural analog of the universal results in [11]. We are also obtain improved endpoint Strichartz estimates for manifolds of nonpositive curvature, which cannot hold for spheres.