On energy-critical half-wave maps into S^2

We consider the energy-critical half-wave maps equation
∂tu+u∧|∇|u=0
for u:[0,T)×R→S2. We give a complete classification of all traveling solitary waves with finite energy. The proof is based on a geometric characterization of these solutions as minimal surfaces with (not necessarily free) boundary on S2. In particular, we discover an explicit Lorentz boost symmetry, which is implemented by the conformal Möbius group on the target S2 applied to half-harmonic maps from R to S2. Complementing our classification result, we carry out a detailed analysis of the linearized operator L around half-harmonic maps Q with arbitrary degree m⩾1 and consisting of m identical Blaschke factors. Here we explicitly determine the nullspace including the zero-energy resonances; in particular, we prove the nondegeneracy of Q. Moreover, we give a full description of the spectrum of L by finding all its L2-eigenvalues and proving their simplicity. Furthermore, we prove a coercivity estimate for L and we rule out embedded eigenvalues inside the essential spectrum. Our spectral analysis is based on a reformulation in terms of certain Jacobi operators (tridiagonal infinite matrices) obtained from a conformal transformation of the spectral problem posed on R to the unit circle S. Finally, we construct a unitary map which can be seen as a gauge transform tailored for a future stability and blowup analysis close to half-harmonic maps. Our spectral results also have potential applications to the half-harmonic map heat flow, which is the parabolic counterpart of the half-wave maps equation.