Near-extreme Reissner-Nordstrom-anti-de Sitter black holes are unstable
against the condensation of an uncharged scalar field with mass close to the
Breitenlohner-Freedman bound. It is shown that a similar instability afflicts
near-extreme large rotating AdS black holes, and near-extreme hyperbolic
Schwarzschild-AdS black holes. The resulting nonlinear hairy black hole
solutions are determined numerically. Some stability results for (possibly
charged) scalar fields in black hole backgrounds are proved. For most of the
extreme black holes we consider, these demonstrate stability if the ``effective
mass" respects the near-horizon BF bound. Small spherical
Reissner-Nordstrom-AdS black holes are an interesting exception to this result.Comment: 34 pages; 13 figure
