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Static solutions of the Einstein-Yang-Mills-Higgs system containing extremeblack holes are studied. The field equations imply strong restrictions onboundary values of all fields at the horizon. If the Yang-Mills radial electricfield $E$ is non-zero there, then all fields at the horizon take values in thecentralizer of $E$. For the particular case of SU(3), there are two differentkinds of centralizers: two-dimensional abelian (Cartan subalgebra) andfour-dimensional (su(2)$\times$u(1)) ones. The two-dimensional centralizeradmits only constant fields: even the geometry of the horizon is that ofconstant curvature. If the cosmological constant $\Lambda$ is negative, atwo-surface of any genus is possible; for positive curvature, only sphericallysymmetrical horizons are allowed. For the four-dimensional centralizer, allspherically symmetrical horizons are explicitly given.Comment: 37 pages, LaTeX, preprint BUTP-94/

Gauge symmetry breakdown at the horizon of extreme black holes