Near soliton evolution for equivariant Schroedinger Maps in two spatial dimensions

We consider the Schr"odinger Map equation in $2+1$ dimensions, with values into ${\mathrm{\S }}^2$. 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 ${\dot{H}}^1$. However, in the process of proving this we also show that within the equivariant class $Q$ is stable in a stronger topology $X \subset \dot H^1$.