Global regularity of wave maps III. Large energy from $\R^{1+2}$ to hyperbolic spaces

Abstract

We show that wave maps $\phi$ from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic spaces $\H^m$ are globally smooth in time if the initial data is smooth, conditionally on some reasonable claims concerning the local theory of such wave maps, as well as the self-similar and travelling (or stationary solutions); we will address these claims in the sequels \cite{tao:heatwave2}, \cite{tao:heatwave3}, \cite{tao:heatwave4} to this paper. Following recent work in critical dispersive equations, the strategy is to reduce matters to the study of an \emph{almost periodic} maximal Cauchy development in the energy class. We then repeatedly analyse the stress-energy tensor of this development (as in \cite{tao:forges}) to extract either a self-similar, travelling, or degenerate non-trivial energy class solution to the wave maps equation. We will then rule out such solutions in the sequels to this paper, establishing the desired global regularity result for wave maps.