Near soliton evolution for $2$-equivariant Schr\"odinger Maps in two space dimensions

We consider equivariant solutions for the Schr\"odinger Map equation in $2+1$ dimensions, with values into $\mathbb{S}^2$. Within each equivariance class $m \in \mathbb{Z}$ this admits a lowest energy nontrivial steady state $Q^m$, which extends to a two dimensional family of steady states by scaling and rotation. If $|m| \geq 3$ then these ground states are known to be stable in the energy space $\dot H^1$, whereas instability and even finite time blow-up along the ground state family may occur if $|m| = 1$. In this article we consider the most delicate case $|m| = 2$. Our main result asserts that small $\dot H^1$ perturbations of the ground state $Q^2$ yield global in time solutions, which satisfy global dispersive bounds. Unlike the higher equivariance classes, here we expect solutions to move arbitrarily far along the soliton family; however, we are able to provide a time dependent bound on the growth of the scale modulation parameter. We also show that within the equivariant class the ground state is stable in a slightly stronger topology $X \subset \dot H^1$.