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We show that an energy-critical wave map into the sphere that has energy just above the degree one harmonic maps and that blows up in finite time asymptotically decouples into a regular part plus a traveling wave with small momentum, in the energy space. In particular, the only possible form of energy concentration is through the concentration of traveling waves. This is often called “quantization of energy” at blow up. The main new tool is a channel of energy type inequality for outgoing small energy wave maps similar to the one proved in our previous work [ 14 ] on outgoing solutions of the energy-critical wave equation. We also give a brief review of important background results in the subcritical and critical regularity theory for the two dimensional wave maps from [ 35 – 37 ,  47 ,  48 ,  51 ,  56 ,  57 ].

Universality of Blow up Profile for Small Blow up Solutions to the Energy Critical Wave Map Equation