Global regularity of wave maps II. Small energy in two dimensions

We show that wave maps from Minkowski space R1+n to a sphere Sm-1 are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space H-n/2 in all dimensions n > 2. This generalizes the results in the prequel [40] of this paper, which addressed the high-dimensional case n greater than or equal to 5. In particular, in two dimensions we have global regularity whenever the energy is small, and global regularity for large data is thus reduced to demonstrating non-concentration of energy.