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We prove global well-posedness and scattering in $H^1$ for the defocusingnonlinear Schr\"{o}dinger equations \begin{equation*} \begin{cases}&(i\partial_t+\Delta_\g)u=u|u|^{2\sigma}; &u(0)=\phi, \end{cases}\end{equation*} on the hyperbolic spaces $\H^d$, $d\geq 2$, for exponents$\sigma\in(0,2/(d-2))$. The main unexpected conclusion is scattering to linearsolutions in the case of small exponents $\sigma$; for comparison, on Euclideanspaces scattering in $H^1$ is not known for any exponent $\sigma\in(1/d,2/d]$and is known to fail for $\sigma\in(0,1/d]$. Our main ingredients are certainnoneuclidean global in time Strichartz estimates and noneuclidean Morawetzinequalities.

Semilinear Schr\"odinger Flows on Hyperbolic Spaces: Scattering in H^1