Sharp Strichartz estimates on nontrapping asymptotically conic manifolds

We obtain the Strichartz inequalities ║u║_{L^qtL^rx ([0,1]×M)} ≥ C║u(0)_L²║(M) for any smooth n-dimensional Riemannian manifold M which is asymptotically conic at infinity (with either short-range or long-range metric perturbation) and nontrapping, where u is a solution to the Schrödinger equation iu_t + 1/2 Δ_Mu = 0, and 2 < q, r ≥ ∞ are admissible Strichartz exponents (2/q + n/r = n/2). This corresponds with the estimates available for Euclidean space (except for the endpoint (q, r) = (2, 2n/n-2) when n > 2). These estimates imply existence theorems for semi-linear Schrödinger equations on M, by adapting arguments from Cazenave and Weissler and Kato.

This result improves on our previous result, which was an L⁴_t,x Strichartz estimate in three dimensions. It is closely related to results of Staffilani-Tataru, Burq, Robbiano-Zuily and Tataru, who consider the case of asymptotically flat manifolds.