Uniqueness of Two-Bubble Wave Maps in High Equivariance Classes

This is the second part of a two-paper series that establishes the uniqueness and regularity of a threshold energy wave map that does not scatter in both time directions. Consider the (Formula presented.) -valued equivariant energy critical wave maps equation on (Formula presented.), with equivariance class (Formula presented.). It is known that every topologically trivial wave map with energy less than twice that of the unique k-equivariant harmonic map (Formula presented.) scatters in both time directions. We study maps with precisely the threshold energy (Formula presented.). In the first part of the series [15] we gave a refined construction of a threshold wave map that asymptotically decouples into a superposition of two harmonic maps (bubbles), one of which is concentrating in scale. In this paper, we show that this solution is the unique (up to the natural invariances of the equation) two-bubble wave map. Combined with our earlier work [14] we obtain an exact description of every threshold wave map.