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

Abstract

We show that wave maps from Minkowski space $\R^{1+n}$ to a sphere $S^{m-1}$ are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space $\dot H^{n/2}$, in all dimensions $n \geq 2$. This generalizes the results in the prequel [math.AP/0010068] of this paper, which addressed the high-dimensional case $n \geq 5$. 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.