Sharp Strichartz Estimates on Non-trapping Asymptotically Conic Manifolds

We obtain the Strichartz inequalities Lt Lx([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 non-trapping, where u is a solution to the Schrödinger equation iut + 1 2 ∆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 [4] and Kato [14]. This result improves on our previous result in [10], which was an Lt,x Strichartz estimate in three dimensions. It is closely related to the results in [22], [1], [26], [19], which consider the case of asymptotically flat manifolds.