Global regularity of wave maps I. Small critical Sobolev norm in high dimension

Abstract

We show that wave maps from Minkowski space $R^{1+n}$ to a sphere are globally smooth if the initial data is smooth and has small norm in the critical Sobolev space ${\dot{H}}^{n/2}$ in the high dimensional case $n\ge 5$. A major difficulty, not present in the earlier results, is that the ${\dot{H}}^{n/2}$ norm barely fails to control $L^{\infty }$, potentially causing a logarithmic divergence in the nonlinearity; however, this can be overcome by using co-ordinate frames adapted to the wave map by approximate parallel transport. In the sequel of this paper we address the more interesting two-dimensional case, which is energy-critical.

24 pages, no figures, to appear, IMRN. The continuity argument has been once again simplified, some references added, and more typoes have been eradicated