  The general classification of 3+1-static black hole solutions of the Einstein
equations, with or without matter, is central in General Relativity and
important in geometry. In the realm of S1-symmetric vacuum spacetimes, a recent
classification proved that, without restrictions on the topology or the
asymptotic behavior, black hole solutions can be only of three kinds: (i)
Schwarzschild black holes, (ii) Boost black holes, or (iii)
Myers-Korotkin-Nicolai black holes, each one having its distinct asymptotic and
topological type. In contrast to this, very little is known about the general
classification of S1-symmetric static electrovacuum black holes although
examples show that, on the large picture, there should be striking differences
with respect to the vacuum case. A basic question then is whether or not there
are charged analogs to the static vacuum black holes of types (i), (ii) and
(iii). In this article we prove the remarkable fact that, while one can
`charge' the Schwarzschild solution (resulting in a Reissner-Nordstr\"om
spacetime) preserving the asymptotic, one cannot do the same to the Boosts and
to the Myers-Korotkin-Nicolai solutions: the addition of a small or large
electric charge, if possible at all, would transform entirely their asymptotic
behavior. In particular, such vacuum solutions cannot be electromagnetically
perturbed. The results of this paper are consistent but go far beyond the works
of Karlovini and Von Unge on periodic analogs of the Reissner-Nordstr\"om black
holes.
  The type of result as well as the techniques used are based on comparison
geometry a la Bakry-\'Emery and appear to be entirely novel in this context.
The findings point to a complex interplay between asymptotic, topology and
charge in spacetime dimension 3+1, markedly different from what occurs in
higher dimensions.
