A refined threshold theorem for (1+2)-dimensional wave maps into surfaces

The recently established threshold theorem for energy critical wave maps states that wave maps with energy less than that of the ground state (i.e., a minimal energy nontrivial harmonic map) are globally regular and scatter on (1+2)-dimensional Minkowski space. In this note we give a refinement of this theorem when the target is a closed orientable surface by taking into account an additional invariant of the problem, namely the topological degree. We show that the sharp energy threshold for global regularity and scattering is in fact twice the energy of the ground state for wave maps with degree zero, whereas wave maps with nonzero degree necessarily have at least the energy of the ground state. We also give a discussion on the formulation of a refined threshold conjecture for the energy critical $SU(2)$ Yang-Mills equation on (1+4)-dimensional Minkowski space.