Harmonic maps with prescribed singularities on unbounded domains

The Einstein/Abelian-Yang-Mills Equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities φ: ℝ³\Σ → ℍ^k+1_ℂ into the (k + 1)-dimensional complex hyperbolic space. In this paper, we prove the existence and uniqueness of harmonic maps with prescribed singularities φ: ℝⁿ\Σ → ℍ, where Σ is an unbounded smooth closed submanifold of ℝⁿ of codimension at least 2, and ℍ is a real, complex, or quaternionic hyperbolic space. As a corollary, we prove the existence of solutions to the reduced stationary and axially symmetric Einstein/Abelian-Yang-Mills Equations.