This timely review provides a self-contained introduction to the mathematical theory of stationary black holes and a self-consistent exposition of the corresponding uniqueness theorems. The opening chapters examine the general properties of space-times admitting Killing fields and derive the Kerr-Newman metric. Strong emphasis is given to the geometrical concepts. The general features of stationary black holes and the laws of black hole mechanics are then reviewed. Critical steps towards the proof of the 'no-hair' theorem are then discussed, including the methods used by Israel, the divergence formulae derived by Carter, Robinson and others, and finally the sigma model identities and the positive mass theorem. The book is rounded off with an extension of the electro-vacuum uniqueness theorem to self-gravitating scalar fields and harmonic mappings. This volume provides a rigorous textbook for graduate students in physics and mathematics. It also offers an invaluable, up-to-date reference for researchers in mathematical physics, general relativity and astrophysics.

The spectrum of known black-hole solutions to the stationary Einstein equations has been steadily increasing, sometimes in unexpected ways. In particular, it has turned out that not all black-hole-equilibrium configurations are characterized by their mass, angular momentum and global charges. Moreover, the high degree of symmetry displayed by vacuum and electro-vacuum black-hole spacetimes ceases to exist in self-gravitating non-linear field theories. This text aims to review some developments in the subject and to discuss them in light of the uniqueness theorem for the Einstein-Maxwell system.

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We present a numerical classification of the spherically symmetric, static solutions to the Einstein-Yang-Mills equations with cosmological constant Λ. We find three qualitatively different classes of configurations, where the solutions in each class are characterized by the value of Λ and the number of nodes, n, of the Yang-Mills amplitude.For sufficiently small, positive values of the cosmological constant, Λ < Λ crit (n), the solutions generalize the Bartnik-McKinnon solitons, which are now surrounded by a cosmological horizon and approach the deSitter geometry in the asymptotic region. For a discrete set of values Λ reg (n) > Λ crit (n), the solutions are topologically 3-spheres, the ground state (n = 1) being the Einstein Universe. In the intermediate region, that is for Λ crit (n) < Λ < Λ reg (n), there exists a discrete family of global solutions with horizon and "finite size".

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.