There has been much progress in recent years in understanding the existence
problem for wave maps with small critical Sobolev norm (in particular for
two-dimensional wave maps with small energy); a key aspect in that theory has
been a renormalization procedure (either a geometric Coulomb gauge, or a
microlocal gauge) which converts the nonlinear term into one closer to that of
a semilinear wave equation. However, both of these renormalization procedures
encounter difficulty if the energy of the solution is large. In this report we
present a different renormalization, based on the harmonic map heat flow, which
works for large energy wave maps from two dimensions to hyperbolic spaces. We
also observe an intriguing estimate of ``non-concentration'' type, which
asserts roughly speaking that if the energy of a wave map concentrates at a
point, then it becomes asymptotically self-similar.Comment: 28 pages, no figures, submitted, Forges les Eaux conference
proceeding
