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

We consider the Schrodinger map initial-value problem ¤ @ t = on R d R; (0) = 0; where : R d R! S 2 ,! 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 2 C(R : H 1 Q ), Q2 S 2 , provided that the data 02 H 1 Q is smooth and satises the smallness condition

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