Harmonic Maps with Prescribed Singularities on Unbounded Domains

TL;DR

This paper proves the existence and uniqueness of harmonic maps with prescribed singularities on unbounded domains, leading to solutions for certain Einstein/Abelian-Yang-Mills equations in symmetric cases.

Contribution

It extends the theory of harmonic maps with prescribed singularities to unbounded domains and different hyperbolic spaces, providing new solutions to Einstein/Abelian-Yang-Mills equations.

Findings

Existence and uniqueness of harmonic maps with prescribed singularities on unbounded domains.

Solutions to stationary, axially symmetric Einstein/Abelian-Yang-Mills equations.

Applicability to real, complex, and quaternionic hyperbolic spaces.

Abstract

The Einstein/Abelian-Yang-Mills Equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities $\p\colon\R^3\sm\Sigma\to\H^{k+1}_\C$ into the $(k+1)$-dimensional complex hyperbolic space. In this paper, we prove the existence and uniqueness of harmonic maps with prescribed singularities $\p\colon\R^n\sm\Sigma\to\H$, where $\Sigma$ is an unbounded smooth closed submanifold of $\R^n$ 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.