Gauge-Symmetry Breakdown at the Horizon of Extreme Black Holes

Static solutions of the Einstein-Yang-Mills-Higgs system containing extreme black holes are studied. The field equations imply strong restrictions on boundary values of all fields at the horizon. If the Yang-Mills radial electric field $E$ is non-zero there, then all fields at the horizon take values in the centralizer of $E$. For the particular case of SU(3), there are two different kinds of centralizers: two-dimensional abelian (Cartan subalgebra) and four-dimensional (su(2)$\times$u(1)) ones. The two-dimensional centralizer admits only constant fields: even the geometry of the horizon is that of constant curvature. If the cosmological constant $\Lambda$ is negative, a two-surface of any genus is possible; for positive curvature, only spherically symmetrical horizons are allowed. For the four-dimensional centralizer, all spherically symmetrical horizons are explicitly given.