On the high-low method for NLS on the hyperbolic space

In this paper, we first prove that the cubic, defocusing nonlinear Schr"odinger equation on the two dimensional hyperbolic space with radial initial data in $H^s({\mathbb{H}}^2)$ is globally well-posed and scatters when $s

\frac{3}{4}$\mathrm{.}Thenweextendtheresulttononlineraitiesoforder$p>3$. The result is proved by extending the high-low method of Bourgain in the hyperbolic setting and by using a Morawetz type estimate proved by the first author and Ionescu.