Harmonic maps to the circle with higher dimensional singular set

TL;DR

This paper constructs harmonic maps from a manifold to the circle with prescribed higher-dimensional singular sets, analyzing their energy and variational properties.

Contribution

It demonstrates how to realize prescribed singular sets as harmonic maps and studies the asymptotic behavior of their energy in variational relaxations.

Findings

Boundary of any oriented submanifold can be realized as a singular set of a harmonic map.

Energy of minimizers converges to the volume of the singular set plus a renormalized interaction energy.

The renormalized energy captures interactions between components of the singular set.

Abstract

In a closed, oriented ambient manifold $(M^n,g)$ we consider the problem of finding $\mathbb{S}^1$-valued harmonic maps with prescribed singular set. We show that the boundary of any oriented $(n-1)$-submanifold can be realised as the singular set of an $\mathbb{S}^1$-valued map, which is classically harmonic away from the singularity and distributionally harmonic across. If the singular set $\Gamma$ is also embedded and $C^{1,1}$, we consider three variational relaxations of the same problem and show that the energy of minimisers converges, after renormalisation, to the volume $\mathcal{H}^{n-2}(\Gamma)$ plus a lower-order "renormalised energy" -- common to all relaxations -- describing an energetic interaction between different components of the singular set.