Concentration compactness for critical wave maps

Abstract

By means of the concentrated compactness method of Bahouri-Gerard and Kenig-Merle, we prove global existence and regularity for wave maps with smooth data and large energy from 2+1 dimensions into the hyperbolic plane. The argument yields an apriori bound of the Coulomb gauged derivative components of our wave map relative to a suitable norm (which holds the solution) in terms of the energy alone. As a by-product of our argument, we obtain a phase-space decomposition of the gauged derivative components analogous to the one of Bahouri-Gerard.