No-hair theorems for black holes in the Abelian Higgs model

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.