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.Comment: Version 2: improved presentation; no changes to the result. Version
3: corrected an oversight in the historical review. Version 4: version
accepted for publicatio
