Non-Existence of Multiple-Black-Hole Solutions Close to Kerr-Newman

We show that a stationary asymptotically flat electro-vacuum solution of Einstein's equations that is everywhere locally almost isometric to a Kerr-Newman solution cannot admit more than one event horizon. Axial symmetry is not assumed. In particular this implies that the assumption of a single event horizon in Alexakis-Ionescu-Klainerman's proof of perturbative uniqueness of Kerr black holes is in fact unnecessary.