  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.
