We consider equivariant solutions for the Schrödinger map problem from R2+1 to H2 with finite energy and show that they are global in time and scatter.

We consider the Schr\"{o}dinger map initial-value problem in dimension two or greater. We prove that the Schr\"{o}dinger map initial-value problem admits a unique global smooth solution, provided that the initial data is smooth and small in the critical Sobolev space. We prove also that the solution operator extends continuously to the critical Sobolev space.