  Motivated by the study of holographic superconductors, we generalize no-hair
theorems for minimally coupled scalar fields charged under an Abelian gauge
field, in arbitrary dimensions and with arbitrary horizon topology. We first
present a straightforward generalization of no-hair theorems for neutral scalar
hair. We then consider the existence of extremal black holes with scalar hair,
and in the case of horizons with zero or positive curvature, provide a bound on
the mass and charge of the scalar field that are necessary for the scalar hair
to develop.
