Semilinear Schrodinger flows on hyperbolic spaces: scattering in H1

We prove global well-posedness and scattering in H-1 for the defocusing nonlinear Schrodinger equations {(i partial derivative(t) + Delta(g))u = u vertical bar u vertical bar(2 sigma); u(0) = phi, on the hyperbolic spaces H-d, d >= 2, for exponents sigma (0, 2/(d - 2)). The main unexpected conclusion is scattering to linear solutions in the case of small exponents s; for comparison, on Euclidean spaces scattering in H-1 is not known for any exponent sigma is an element of (1/d, 2/d] and is known to fail for sigma is an element of (0, 1/d]. Our main ingredients are certain noneuclidean global in time Strichartz estimates and noneuclidean Morawetz inequalities.