A controlling norm for energy-critical Schrödinger maps

We consider energy-critical Schrödinger maps with target either the sphere S 2 \mathbb {S}^2 or hyperbolic plane H 2 \mathbb {H}^2 and establish that a unique solution may be continued so long as a certain space-time L 4 L^4 norm remains bounded. This reduces the large data global wellposedness problem to that of controlling this norm.