Harmonic maps to the circle with higher dimensional singular set

In a closed, oriented ambient manifold we consider the problem of finding ‐valued harmonic maps with prescribed singular set. We show that the boundary of any oriented ‐submanifold can be realised as the singular set of an ‐valued map, which is classically harmonic away from the singularity and distributionally harmonic across. If the singular set is also embedded and , we consider three variational relaxations of the same problem and show that the energy of minimisers converges, after renormalisation, to the volume plus a lower order ‘renormalised energy’, common to all relaxations, describing an energetic interaction between different components of the singular set.