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We consider the Schr\"odinger Map equation in $2+1$ dimensions, with valuesinto $\S^2$. This admits a lowest energy steady state $Q$, namely thestereographic projection, which extends to a two dimensional family of steadystates by scaling and rotation. We prove that $Q$ is unstable in the energyspace $\dot H^1$. However, in the process of proving this we also show thatwithin the equivariant class $Q$ is stable in a stronger topology $X \subset\dot H^1$.

Near soliton evolution for equivariant Schroedinger Maps in two spatial dimensions