Asymptotic stability conjecture for anti-de Sitter spacetime with dissipative boundary conditions

The pure anti-de Sitter spacetime has metric

$$g_{AdS}=-\left(1+\bigl(-\frac{\Lambda}{3}\bigr)r^2\right)dt^2+\left(1+\bigl(-\frac{\Lambda}{3}\bigr)r^2\right)^{-1}dr^2+r^2d\sigma^2.$$

The boundary conditions are optimally dissipative. Asymptotic stability conjecture. Anti-de Sitter spacetime is asymptotically stable for optimally dissipative boundary conditions. Holzegel, Luk, Smulevici, and Warnick proved standard energy boundedness and integrated decay for the Klein–Gordon equation with $\alpha=-2$ under these boundary conditions, providing a first step toward this conjecture; the full nonlinear stability statement remains open.