We construct blow-up solutions of the energy critical wave map equation on $mathbb{R}^{2+1}to mathcal N$ with polynomial blow-up rate ($t^{-1- u}$ for blow-up at $t=0$) in the case when $mathcal{N}$ is a surface of revolution. Here we extend the blow
-up range found by C^arstea ($ u>frac 12$) based on the work by Krieger, Schlag and Tataru to $ u>0$. This work relies on and generalizes the recent result of Krieger and the author where the target manifold is chosen as the standard sphere.
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 $la
mbda(t) = t^{-1- u}$ 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.
