Linear Stability of Schwarzschild-Anti-de Sitter spacetimes III: Quasimodes and sharp decay of gravitational perturbations

In this last part of the series we prove that the slow (inverse logarithmic) decay in time of solutions to the linearised Einstein equations on Schwarzschild-Anti-de Sitter backgrounds obtained in~\cite{Gra.Hol24,Gra.Hol24a} is in fact optimal by constructing quasimode solutions for the Teukolsky system. The main difficulties compared with the case of the scalar wave equation treated in earlier works arise from the first order terms in the Teukolsky equation, the coupling of the Teukolsky quantities at the conformal boundary and ensuring that the relevant quasimode solutions satisfy the Teukolsky-Starobinsky relations. The proof invokes a quasimode construction for the corresponding Regge-Wheeler system (which can be fully decoupled at the expense of a higher order boundary condition) and a reverse Chandrasekhar transformation which generates solutions of the Teukolsky system from solutions of the Regge-Wheeler system. Finally, we provide a general discussion of the well-posedness theory for the higher order boundary conditions that typically appear in the process of decoupling.