On the existence of smooth self-similar blowup profiles for the wave map equation

Consider the equivariant wave map equation from Minkowski space to a rotationally symmetric manifold N that has an equator (e.g., the sphere). In dimension 3, this paper presents a necessary and sufficient condition on N for the existence of a smooth self-similar blowup profile. More generally, we study the relation between • the minimizing properties of the equator map for the Dirichlet energy corresponding to the (elliptic) harmonic map problem and • the existence of a smooth blowup profile for the (hyperbolic) wave map problem. This has several applications to questions of regularity and uniqueness for the wave map equation.