STATICITY AND UNIQUENESS OF MULTIPLE BLACK-HOLE SOLUTIONS OF SIGMA-MODELS

We prove the staticity and no-hair conjectures for self-gravitating no n-linear sigma-models with Riemannian target manifolds. We first demon strate that any self-coupled, stationary scalar mapping (sigma-model) from a strictly stationary domain of outer communications with non-rot ating horizon to a Riemannian manifold has to be static. Applying the positive mass theorem, we subsequently show that the exterior Schwarzs child geometry is the only maximally extended, static, asymptotically flat solution of the coupled Einstein-sigma-model equations with regul ar (but not necessarily connected) horizon. The line of reasoning in t he second part of the article is adopted from the work of Bunting and Masood-ul-Alam, who proved the uniqueness theorem in the vacuum case.