Optimal polynomial blow up range for critical wave maps

We prove that the critical Wave Maps equation with target $S^2$ and origin $mathbb{R}^{2+1}$ admits energy class blow up solutions of the form $$u(t,r)=Q(lambda(t)r)+epsilon(t,r)$$where $Q: mathbb{R}^2 to S^2$ is the ground state harmonic map and $lambda(t) = t^{-1- u}$ for any $ u > 0$. This extends the work [13], where such solutions were constructed under the assumption $ u > 1/2$. In light of a result of Struwe [22], our result is optimal for polynomial blow up rates.