Minimizers of the [Formula presented]-energy of [Formula presented]-valued maps with prescribed singularities. Do they exist?

The paper is concerned with the least [Formula presented]-energy required to produce maps from a domain [Formula presented] with values into [Formula presented] having prescribed singularities [Formula presented]. The value of the infimum has been known for a long time and corresponds to the length of minimal configurations connecting the points [Formula presented] between themselves and/or to the boundary. We tackle here the question whether the infimum of this [Formula presented]-energy is achieved. This natural topic turns out to be delicate and we have a complete answer only when [Formula presented]. The bottom line for [Formula presented] is that the infimum is “rarely” achieved. As a “substitute”, we give a full description of the asymptotic behavior of all minimizing sequences and show that they “concentrate” along “convex combinations” of minimal configurations.