We study the moduli space of spherically symmetric, asymptotically flat solutions of the Einstein–Maxwell equations coupled to a real scalar field. Near the Reissner–Nordström family, I will present a proof showing that the black hole region is foliated by codimension-one stable manifolds of constant charge-to-mass ratio, and that the black hole formation threshold is exactly the set of asymptotically extremal black holes. This is based on upcoming joint work with Y. Angelopoulos (BIMSA) and R. Unger (Berkeley).