### Abstract

We obtain the Strichartz inequalities ∥u∥ _{LtqLxr([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}, + 1/2Δ _{M}u = 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 L _{t,x} ^{4} 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.

Original language | English (US) |
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Pages (from-to) | 963-1024 |

Number of pages | 62 |

Journal | American Journal of Mathematics |

Volume | 128 |

Issue number | 4 |

DOIs | |

State | Published - Aug 1 2006 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*American Journal of Mathematics*,

*128*(4), 963-1024. https://doi.org/10.1353/ajm.2006.0033