Abstract
We prove global well-posedness and scattering in H 1 for the defocusing nonlinear Schrödinger equations $$\left\{\begin{array}{ll}(i\partial_t+\Delta_g)u=u|u|^{2\sigma};\\u(0)=\phi,\end{array}\right.$$ on the hyperbolic spaces $${\mathbb{H}^d}$$ , d ≥ 2, for exponents $${\sigma \in (0, 2/(d-2))}$$ . The main unexpected conclusion is scattering to linear solutions in the case of small exponents σ; for comparison, on Euclidean spaces 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 certain noneuclidean global in time Strichartz estimates and noneuclidean Morawetz inequalities.
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