  Motivated by stationary vacuum solutions of the Einstein field equations, we
study singular harmonic maps from domains of 3-dimensional Euclidean space to
the hyperbolic plane having bounded hyperbolic distance to Kerr harmonic maps.
In the degenerate case, we prove that every such harmonic map admits a unique
tangent harmonic map at the extreme black hole horizon. The possible tangent
maps are classified and shown to be shifted 'extreme Kerr' geodesics in the
hyperbolic plane that depend on two parameters, one determined by angular
momentum and another by conical singularities. In addition, rates of
convergence to the tangent map are established. Similarly, expansions in the
asymptotically flat end are presented. These results, together with those of
Li-Tian and Weinstein, provide a complete regularity theory for harmonic maps
from $\mathbb R^3\setminus z\text{-axis}$ to $\mathbb H^2$ with prescribed
singularities. Lastly, the analysis is utilized to prove existence of the so
called near horizon limit, and to compute the associated near horizon
geometries of extreme black holes.
