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 et al. (Ginzburg–Landau vortices, progress in nonlinear differential equations and their applications, vol 13, Birkhäuser, Boston, 1994) for the circle. 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.

我们为映射定义了归一化能量，该归一化能量描述了由于边界数据和目标流形的多重连接所产生的障碍而导致的奇异性以外的谐波映射的一阶渐近性。这些构造概括了Bethuel等人的定义。（金茨堡-兰道涡，非线性微分方程的进展及其应用，第13卷，伯克希尔，波士顿，1994年）。通常，奇异点是几何对象，可以通过新的谐调概念研究对同位异点的依赖性。重新归一化的能量显示为矫顽力和Lipschitz连续的。重新归一化的能量与最小化可重新归一化的奇异谐波图相关，并且最小化点的配置可以通过奇异点处的应力-能量张量的通量来表征。我们在几种特定情况下计算奇异能量和重新规范化的能量。