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.