Gravitational mode stability and the nonlinear stability of Kerr–de Sitter black holes in the full subextremal range

We prove that subextremal Kerr–de Sitter black holes are nonlinearly stable as solutions of the Einstein vacuum equations with positive cosmological constant, in the full subextremal range. Hintz, Petersen and Vasy reduced this to mode stability of the ungauged linearized Einstein equation; we prove that statement. The main step is mode stability of the Teukolsky equations of spin ±2: there are no nontrivial outgoing modes with Im σ ≥ 0. From it we deduce that every generalized mode of the linearized Einstein equation with Im σ ≥ 0 is pure gauge, up to a linearized Kerr–de Sitter metric when σ = 0. Part of the proof is computer-assisted. This work was developed with substantial AI assistance (see the Statement on the use of AI in the paper) and has not yet been checked by a human expert.