Stability of Gauss-Bonnet black holes in Anti-de-Sitter space-time against scalar field condensation

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We study the stability of static, hyperbolic Gauss-Bonnet black holes in (4+1)-dimensional Anti-de-Sitter (AdS) space-time against the formation of scalar hair.

• Close to extremality the black holes possess a near-horizon topology of AdS_2 x H^3 such that within a certain range of the scalar field mass one would expect that they become unstable to the condensation of an uncharged scalar field.
• We confirm this numerically and observe that there exists a family of hairy black hole solutions labelled by the number of nodes of the scalar field function.
• We construct explicit examples of solutions with a scalar field that possesses zero nodes, one node and two nodes, respectively, and show that the solutions with nodes persist in the limit of Einstein gravity, i.e.