Full blow-up range for co-rotaional wave maps to surfaces of revolution

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.