Stability of the Toroidal AdS Schwarzschild Solution in the Einstein--Klein-Gordon System

We consider the stability of the toroidal AdS-Schwarzshild black holes as solutions of the Einstein–Klein-Gordon system, with Dirichlet or Neumann boundary conditions for the scalar field. Restricting to perturbations that respect the toroidal symmetry we show both orbital and asymptotic stability for the full nonlinear problem, for a range of choices of the Klein-Gordon mass. The solutions we construct with Neumann boundary conditions have a Hawking mass which diverges towards infinity, reflecting the infinite energy of the Klein-Gordon field for perturbations satisfying these boundary conditions.