Radiation Fields on Asymptotically Euclidean Manifolds

Radiation Fields, Scattering and Inverse Scattering on Asymptotically Hyperbolic Manifolds

We define the forward and backward radiation fields on an asymptotically hyperbolic manifold and show that they give unitary translation representations of the wave group, and as such can be used to define a scattering matrix. We show that this scattering matrix is equivalent to the one defined by stationary methods. Furthermore, we prove a support theorem for the radiation fields which general...

Distribution of Resonances for Asymptotically Euclidean Manifolds

The Semilinear Wave Equation on Asymptotically Euclidean Manifolds

We consider the quadratically semilinear wave equation on (R, g), d ≥ 3. The metric g is non-trapping and approaches the Euclidean metric like 〈x〉. Using Mourre estimates and the Kato theory of smoothness, we obtain, for ρ > 0, a Keel–Smith–Sogge type inequality for the linear equation. Thanks to this estimate, we prove long time existence for the nonlinear problem with small initial data for ρ...

Concerning the Wave Equation on Asymptotically Euclidean Manifolds

We obtain KSS, Strichartz and certain weighted Strichartz estimate for the wave equation on (R, g), d ≥ 3, when metric g is non-trapping and approaches the Euclidean metric like 〈x〉 with ρ > 0. Using the KSS estimate, we prove almost global existence for quadratically semilinear wave equations with small initial data for ρ > 1 and d = 3. Also, we establish the Strauss conjecture when the metric...

Ricci-Flat Anti-Self-Dual Asymptotically Locally Euclidean 4-Manifolds

of the Dissertation Ricci-Flat Anti-Self-Dual Asymptotically Locally Euclidean 4-Manifolds by Evan Patrick Wright Doctor of Philosophy in Mathematics Stony Brook University 2013 A classification result for Ricci-flat anti-self-dual asymptotically locally Euclidean 4-manifolds is obtained: they are either hyperkähler (one of the gravitational instantons classified by Kronheimer), or a cyclic quo...