Global regularity of critical Schr\"odinger maps: subthreshold dispersed energy

We consider the energy-critical Schroedinger map initial value problem with smooth initial data from R^2 into the sphere S^2. Given sufficiently energy-dispersed data with subthreshold energy, we prove that the system admits a unique global smooth solution. This improves earlier analogous conditional results. The key behind this improvement lies in exploiting estimates on the commutator of the Schroedinger map and harmonic map heat flows.