The authors 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. The authors prove that Q is unstable in the energy space ?¹. However, in the process of proving this they also show that within the equivariant class Q is stable in a stronger topology XC?¹.