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We prove Birkhoff-type results showing that $L^2$ solutions of the linearizedEinstein equations around Riemannian Kottler ("Schwarzschild-anti de Sitter")metrics in arbitrary dimension and horizon topology, which are not controlledby "master functions" are pure gauge. Together with earlier results thisimplies that the $TT$-gauge-fixed linearized Einstein operator for thesemetrics is non-degenerate for open ranges of the mass parameter.

Non-degeneracy of Riemannian Schwarzschild-anti de Sitter metrics: Birkhoff-type results in linearized gravity