Global Nonlinear Stability of the Kerr-de Sitter Family for Small Angular Momentum in General Relativity

We present a rigorous proof of the global nonlinear stability of the Kerr-de Sitter family of black holes within the context of the initial value problem for the Einstein vacuum equations with a positive cosmological constant > 0. We consider initial data sets (, h, K) that are small perturbations of the data induced by a slowly rotating Kerr-de Sitter spacetime with parameters (M0, a0) satisfying |a0| M0. By employing a generalized harmonic gauge with constraint damping and utilizing microlocal analysis techniques to resolve the normally hyperbolic trapping at the photon sphere, we establish that the maximal globally hyperbolic development of such data is future causally geodesically complete. Furthermore, we prove that the solution decays exponentially in time to a member of the Kerr-de Sitter family with perturbed parameters (M, a) near (M0, a0). The analysis relies crucially on the presence of a spectral gap for the linearized operator, which circumvents the slow polynomial decay characteristic of the asymptotically flat =0 setting.