Harmonic maps with prescribed singularities into Hadamard manifolds

Let M a Riemannian manifold of dimension m ≥ 3, let Σ be a closed smooth submanifold of M of co-dimension at least 2, and let H be a Hadamard manifold with pinched sectional curvatures. We prove the existence and uniqueness of harmonic maps φ. M \ Σ → H with prescribed singularities along Σ. When M = ℝ³, and H = H^k_ℂ, the complex hyperbolic space, this result has applications to the problem of multiple co-axially rotating black holes in general relativity.