On the Dirichlet problem for harmonic maps with prescribed singularities

TL;DR

This paper establishes existence and uniqueness results for harmonic maps with prescribed singularities into rank-one symmetric spaces, extending previous work and with applications to black hole configurations in General Relativity.

Contribution

It generalizes earlier results on harmonic maps into hyperbolic spaces to broader symmetric spaces, addressing the Dirichlet problem with singularities.

Findings

Proves existence and uniqueness of harmonic maps with prescribed singularities.

Extends previous results from hyperbolic plane to general rank-one symmetric spaces.

Applies to models of rotating charged black holes in General Relativity.

Abstract

Let $\M$ be a classical Riemannian globally symmetric space of rank one and non-compact type. We prove the existence and uniqueness of solutions to the Dirichlet problem for harmonic maps into $\M$ with prescribed singularities along a closed submanifold of the domain. This generalizes our previous work where such maps into the hyperbolic plane were constructed. This problem, in the case where $\M$ is the complex-hyperbolic plane, has applications to equilibrium configurations of co-axially rotating charged black holes in General Relativity.