Asymptotic Analysis of Harmonic Maps With Prescribed Singularities

This is the first in a series of two papers to establish the mass-angular momentum inequality for multiple black holes. We study singular harmonic maps from domains of 3-dimensional Euclidean space to the hyperbolic plane having bounded hyperbolic distance to extreme Kerr harmonic maps. 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 [24, 25] and Weinstein [35,36], provide a complete regularity theory for harmonic maps from $\mathbb R^3\setminus z\text{-axis}$ to $\mathbb H^2$ with these prescribed singularities. The analysis is additionally utilized to prove existence of the so called near horizon limit, and to compute the associated near horizon geometries of extreme black holes.