G. Weinstein

Title: Multi-black-hole configurations

Abstract: In order to complete the proof of uniqueness of the Kerr solutions among stationary axisymmetric solutions of the Einstein Vacuum Equations, it is necessary to rule out configurations with disconnected horizons. The problem of finding all globally hyperbolic, asymptotically flat, stationary axisymmetric solutions of the Einstein Vacuum Equations with non-degenerate horizons is reduced to a singular boundary value problem for a harmonic map from a domain in R^3 into the hyperbolic plane H^2. Generalizations to the Einstein Maxwell, and Einstein Abelian Yang-Mills are also considered leading to similar problems into the complex hyperbolic spaces H^k+1_C. An existence and uniqueness theorem for such harmonic maps with singularities is proved, and applied to the multi black hole problem. The obstruction to reconstructing the spacetime solution from a solution of the reduced problem is discussed. This obstruction appears as a conical singularity on the axis between any two black holes and can be interpreted as a force acting between the black holes.

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Copyright 29.VIII.02 by P.T.Chrusciel, A.Chopin, and G. Weinstein