Stability of the expanding region of Kerr de Sitter spacetimes

We prove the nonlinear stability of the cosmological region of Kerr de Sitter spacetimes. More precisely, we show that solutions to the Einstein vacuum equations with positive cosmological constant arising from data on a cylinder that is uniformly close to the Kerr de Sitter geometry (with possibly different mass and angular momentum parameters at either end) are future geodesically complete and display asymptotically de Sitter-like degrees of freedom. The proof uses an ADM formulation of the Einstein equations in parabolic gauge. Together with a well-known theorem of Hintz-Vasy [Acta Math. 220 (2018)], our result yields a global stability result for Kerr de Sitter from Cauchy data on a spacelike hypersurface bridging two black hole exteriors.