Boundedness and growth for the massive wave equation on asymptotically anti-de Sitter black holes

We study the global dynamics of free massive scalar fields on general, globally stationary, asymptotically AdS black hole backgrounds with Dirichlet, Neumann or Robin boundary conditions imposed on ψ   at infinity. This class includes the regular Kerr–AdS black holes satisfying the Hawking–Reall bound r+2>|a|l. We establish a suitable criterion for linear stability (in the sense of uniform boundedness) of ψ and demonstrate how the issue of stability can depend on the boundary condition prescribed. In particular, in the slowly rotating Kerr–AdS case, we obtain the existence of linear scalar hair (i.e. non-trivial stationary solutions) for suitably chosen Robin boundary conditions.