  Using the harmonic map heat flow, we construct an energy class for wave maps
$\phi$ from two-dimensional Minkowski space $\R^{1+2}$ to hyperbolic spaces
$\H^m$, and then show (conditionally on a large data well-posedness claim for
such wave maps) that no stationary, travelling, self-similar, or degenerate
wave maps exist in this energy class. These results form three of the five
claims required in our earlier paper (arXiv:0805.4666) to prove global
regularity for such wave maps. (The conditional claim of large data
well-posedness is one of the remaining claims required in that paper.)
