Global Schrodinger maps in dimensions d ≥ 2: Small data in the critical Sobolev spaces

We consider the Schrodinger map initial-value problem {partial derivative t phi - phi x Delta phi on R-d x R, phi(0) = phi(0) where phi: Rd x R -> R-2 hooked right arrow R-3 is a smooth function. In all dimensions d >= 2, we prove that the Schrodinger map initial-value problem admits a unique global smooth solution phi is an element of C (R : H-Q(infinity)), Q is an element of S-2 Q), Q 2 S 2, provided that the data phi(0) is an element of H-Q(infinity) is smooth and satisfies the smallness condition vertical bar vertical bar phi(0) - Q vertical bar vertical bar(Hd/2) << 1. We prove also that the solution operator extends continuously to the space of data in Hd/2 boolean AND H-Q(d/2-1) with small H-d/2 norm.