Coulomb Sectors and Scattering for Maxwell-Higgs Fields on Schwarzschild and Slowly Rotating Kerr Backgrounds

We develop a small-data Maxwell--Higgs theory on Schwarzschild and slowly rotating Kerr black-hole exteriors for gauge-invariant nonnegative self-interactions near the trivial vacuum. The Schwarzschild part gives a complete global, radiative, and scattering theory, while the slowly rotating Kerr part gives a robust massless forward theory and a perturbative small-electric extension. The main mechanism is a transfer principle: once the required linear energy, decay, horizon, and far-field estimates are available, the nonlinear Lorenz-gauge problem yields global existence, gauge-covariant radiation fields, nonlinear wave operators, and asymptotic completeness. The Coulomb-sector analysis identifies the correct long-range normalization in fixed electric sectors and separates the genuinely proved results from the remaining rotating massive final-state problems. All Kerr scattering statements beyond the established massless and small-electric forward regimes are stated explicitly under their necessary spectral and final-state conditions, namely, no rapid-rotation, large-charge, and unconditional massive rotating scattering.