Near Sharp Strichartz estimates with loss in the presence of degenerate hyperbolic trapping

We consider an $n$-dimensional spherically symmetric, asymptotically Euclidean manifold with two ends and a codimension 1 trapped set which is degenerately hyperbolic. By separating variables and constructing a semiclassical parametrix for a time scale polynomially beyond Ehrenfest time, we show that solutions to the linear Schr\"odiner equation with initial conditions localized on a spherical harmonic satisfy Strichartz estimates with a loss depending only on the dimension $n$ and independent of the degeneracy. The Strichartz estimates are sharp up to an arbitrary $\beta>0$ loss. This is in contrast to \cite{ChWu-lsm}, where it is shown that solutions satisfy a sharp local smoothing estimate with loss depending only on the degeneracy of the trapped set, independent of the dimension

Existence and stability of solitons for the nonlinear Schr\"odinger equation on hyperbolic space

We study the existence and stability of ground state solutions or solitons to a nonlinear stationary equation on hyperbolic space. The method of concentration compactness applies and shows that the results correlate strongly to those of Euclidean space.Comment: New: As noted in Banica-Duyckaerts (arXiv:1411.0846), Section 5 should read that for sufficiently large mass, sub-critical problems can be solved via energy minimization for all d \geq 2 and as a result Cazenave-Lions results can be applied in Section 6 with the same restriction. These requirements were addressed by the subsequent work with Metcalfe and Taylor in arXiv:1203.361