A no-hair theorem for self-gravitating nonlinear sigma models

The coupled system of gravity and mappings φ:(M,g)→(N,G) with harmonic action and additional potential is considered. For spherically symmetric manifolds (M,g) and Riemannian manifolds (N,G) it is shown that the only static, asymptotically flat solutions of the coupled Einstein-matter equations with regular event horizon and finite energy consist of the Schwarzschild metric and a constant map, being a zero of the non-negative potential.