The nonlinear Schr\"odinger equation on the hyperbolic space

In this article we study some aspects of dispersive and concentration phenomena for the Schr"odinger equation posed on hyperbolic space ${\mathbb{H}}^n$, in order to see if the negative curvature of the manifold gets the dynamics more stable than in the Euclidean case. It is indeed the case for the dispersive properties : we prove that the dispersion inequality is valid, in a stronger form than the one on ${\mathbb{R}}^n$. However, the geometry does not have enough of an effect to avoid the concentration phenomena and the picture is actually worse than expected. The critical nonlinearity power for blow-up turns out to be the same as in the euclidean case, and we prove that there are more explosive solutions for critical and supercritical nonlinearities.