Optimal polynomial blow up range for critical wave maps

We prove that the critical Wave Maps equation with target $S^2$ and origin $\R^{2+1}$ admits energy class blow up solutions of the form \[ u(t, r) = Q(\lambda(t)r) + \eps(t, r) \] where $Q:\R^2\rightarrow S^2$ is the ground state harmonic map and $\lambda(t) = t^{-1-\nu}$ for any $\nu>0$. This extends the work $\cite{KST0}$, where such solutions were constructed under the assumption $\nu>\frac{1}{2}$. In light of a result of Struwe $\cite{Struwe1}$, our result is optimal for polynomial blow up rates.