Remarks on local theory for Schr\"odinger maps near harmonic maps

We consider the initial-value problem for the equivariant Schr\"odinger maps near a family of harmonic maps. We provide some supplemental arguments for the proof of local well-posedness result by Gustafson, Kang and Tsai in [Duke Math. J. 145(3) 537--583, 2008]. We also prove that the solution near harmonic maps is unique in $C(I;\dot{H}^1(\mathbb{R}^2)\cap\dot{H}^2(\mathbb{R}^2))$ for time interval $I$. In the proof, we give a justification of the derivation of the modified Schr\"odinger map equation in low regularity settings without smallness of energy.