On Schrödinger maps

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An Approximation Scheme for Schrödinger Maps (2007, background): “For the case of maps from 2 to either the sphere 2 or the hyperbolic space 2, Nahmod, Stefanov, and Uhlenbeck proved that solutions (to a modified version of the Schrödinger map problem) exist in H2+ 2 for any > 0 (Nahmod et al., 2003, 2004).”
Global existence and uniqueness of Schrödinger maps in dimensions d... 4 (2007, methodology): “Because of this, the stereographic model used in [6], [2], and [7] is not relevant, and it does not appear possible to prove Theorem 1.”
On the global well-posedness of the one-dimensional Schrodinger map flow (2008, background): “Thus we have reduced ourselves to the same situation as in [Chang et al. 2000] (the only difference in (24) from the case where the target is a Riemann surface is that the equation for each 8 j depends also on the other 8k, k = 1, .”
Low-regularity Schrödinger maps, II: global well-posedness in dimensions d ≥ 3 (2007, methodology): “At about the same time and independently, Bejenaru [1] proved local well-posedness of the initial-value problem (1.6) for small data in the Sobolev spaces H σ , for σ in the full subcritical range σ > d/2.”