Nonlinear instability and asymptotic stability of anti-de Sitter spacetime

Let anti-de Sitter spacetime be the maximally symmetric solution of the vacuum Einstein equations with negative cosmological constant. Consider perturbations of anti-de Sitter spacetime under reflecting boundary conditions, for which the curvature energy flux through the timelike boundary vanishes, and under optimally dissipative boundary conditions, for which this flux is as large as possible. Anti-de Sitter instability–stability conjecture. Anti-de Sitter spacetime is non-linearly unstable for reflecting boundary conditions and asymptotically stable for optimally dissipative boundary conditions. This conjecture concerns the global dynamics of perturbations in the Einstein equations with negative cosmological constant. The appropriate geometric boundary conditions and the resulting global behavior remain largely unknown, while Friedrich's work provides local-in-time solutions for such problems.