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

We consider the Schrödinger map initial-value problem, where φR{double-struck}^d × R{double-struck} → S{double-struck}² → R{double-struck}³ is a smooth function. In all dimensions d ≥ 2, we prove that the Schrödinger map initial-value problem admits a unique global smooth solution φ ε C(R{double-struck} : H_Q^∞), Q ε S{double-struck}², provided that the data φo ε H_Q^∞ is smooth and satisfies the smallness condition ||φ0 - Q||_H^d/2 << 1. We prove also that the solution operator extends continuously to the space of data in H.^d/2 ∩ H_Q ^d/2-1 with small H ^d/2. norm.