Despite the recent evidence that anti-de Sitter (AdS) spacetime is nonlinearly unstable, we argue that many asymptotically AdS solutions are nonlinearly stable. This includes geons, boson stars and black holes. As part of our argument, we calculate the frequencies of long-lived gravitational quasinormal modes of AdS black holes in various dimensions. We also discuss a new class of asymptotically AdS solutions describing noncoalescing black hole binaries.
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