Schrödinger flow near harmonic maps

For the Schrödinger flow from ℝ² × ℝ⁺ to the 2-sphere S², it is not known if finite energy solutions can blow up in finite time. We study equivariant solutions whose energy is near the energy of the family of equivariant harmonic maps. We prove that such solutions remain close to the harmonic maps until the blowup time (if any), and that they blow up if and only if the length scale of the nearest harmonic map goes to 0.