Charged parallel spinors and applications to mass--charge inequalities

We investigate the equality case of the spin positive mass theorem with charge in the Riemannian setting. This leads naturally to the notion of charged parallel spinor, which plays a central role in the analysis of extremal charged manifolds. As an application, we characterize the equality case of the mass--charge inequality in terms of the extremal Reissner--Nordstr\"om geometry for asymptotically flat manifolds with connected boundary and for manifolds with a single asymptotically cylindrical end.