  Consider the equivariant wave map equation from Minkowski space to a
rotationnally symmetric manifold which has an equator (example: the sphere). In
dimension 3, this article gives a necessary and sufficient condition for the
existence of a smooth self-similar blow up profile. More generally, we study
the relation between 1. the minimizing properties of the equator map for the
(elliptic) Dirichlet energy and 2. the existence of a smooth blow-up profile
for the (hyperbolic) wave map problem. Several applications of this approach
are described.
