Long time solutions for wave maps with large data

For 2 + 1 dimensional wave maps with ${\mathbb{S}}^2$ as the target, we show that for all positive numbers $T_0>0$ and $E_0>0$, there exist Cauchy initial data with energy at least $E_0$, so that the solution’s life-span is at least $[0,T_0]$. We assume neither symmetry nor closeness to harmonic maps.