Scattering theory for radial nonlinear Schr\"{o}dinger equations on   hyperbolic space
  We study the long time behavior of radial solutions to nonlinear Schr\"{o}dinger equations on hyperbolic space. We show that the usual distinction between short range and long range nonlinearity is modified: the geometry of the hyperbolic space makes every power-like nonlinearity short range. The proofs rely on weighted Strichartz estimates, which imply Strichartz estimates for a broader family of admissible pairs, and on Morawetz type inequalities. The latter are established without symmetry assumptions. 
Valeria Banica (DP), R\'emi Carles (MAB), Gigliola Staffilani (MIT)