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

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