Strichartz estimates for the Schrödinger equation on negatively curved compact manifolds

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