We consider the Schrödinger Map equation in 2 + 1 dimensions, with values into S2. This admits a lowest energy steady state Q, namely the stereographic projection, which extends to a two dimensional family of steady states by scaling and rotation. We prove that Q is unstable in the energy space H1. However, in the process of proving this we also show that within the equivariant class Q is stable in a stronger topology X ⊂ H 1. © 2013 by the American Mathematical Society. All rights reserved.

We consider equivariant solutions for the Schrödinger map problem from ℝ2+1 to S2 with energy less than 4π and show that they are global in time and scatter.