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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 forsuch wave maps) that no stationary, travelling, self-similar, or degeneratewave maps exist in this energy class. These results form three of the fiveclaims required in our earlier paper (arXiv:0805.4666) to prove globalregularity for such wave maps. (The conditional claim of large datawell-posedness is one of the remaining claims required in that paper.)

Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class