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Title: Are Kerr-AdS black holes nonlinearly stable with reflecting boundary conditions?
Plain: In anti-de Sitter space, a model universe with a reflecting boundary, waves cannot escape and may concentrate into ever smaller scales. Even empty anti-de Sitter space is unstable, and the fate of its black holes is unknown.
Precise: For Kerr-AdS with reflecting boundary conditions at conformal infinity, linear waves decay only logarithmically in time (Holzegel and Smulevici), and superradiant instabilities occur when $\Omega_{H}\ell > 1$, with $\Omega_{H}$ the horizon angular velocity and l the AdS radius. Determine whether small perturbations of Kerr-AdS with $\Omega_{H}\ell < 1$ remain small for all time or develop turbulent cascades, and identify the end state when $\Omega_{H}\ell > 1$. An answer is a theorem or convergent numerical evidence.
Settled by: A nonlinear stability or instability theorem for Kerr-AdS, or long-time convergent simulations resolving the cascade.
Kind: well-posed
Answer type: yes-no
References: (empty)
Literature status: open
Status note: Pure AdS is nonlinearly unstable (Bizon and Rostworowski 2011 numerics; proved for Einstein-null dust by Moschidis 2017-2018); Kerr-AdS is open as of 2026.
Posed since: (empty)