Renormalised energies and renormalisable singular harmonic maps into a compact manifold on planar domains

TL;DR

This paper introduces a generalized framework for defining and analyzing renormalised energies of harmonic maps with singularities on planar domains, extending previous models and exploring their geometric and variational properties.

Contribution

It generalizes the concept of renormalised energies for harmonic maps to more complex target manifolds and boundary conditions, introducing new notions like synharmony and studying their properties.

Findings

Renormalised energies are coercive and Lipschitz-continuous.

Singularities are characterized as geometrical objects with flux-based descriptions.

Explicit computations of energies in specific cases are provided.

Abstract

We define renormalised energies for maps that describe the first-order asymptotics of harmonic maps outside of singularities arising due to obstructions generated by the boundary data and the mutliple connectedness of the target manifold. The constructions generalise the definition by Bethuel, Brezis and H\'elein for the circle (Ginzburg-Landau vortices, 1994). In general, the singularities are geometrical objects and the dependence on homotopic singularities can be studied through a new notion of synharmony. The renormalised energies are showed to be coercive and Lipschitz-continuous. The renormalised energies are associated to minimising renormalisable singular harmonic maps and minimising configurations of points can be characterised by the flux of the stress-energy tensor at the singularities. We compute the singular energy and the renormalised energy in several particular cases.