The Maxwell-Higgs System with Scalar Potential on Subextremal Kerr Spacetimes: Nonlinear wave operators and asymptotic completeness

TL;DR

This paper develops a mathematical framework for understanding the scattering and asymptotic behavior of the Maxwell-Higgs system on Kerr black hole spacetimes, establishing small-data completeness and detailed scattering maps.

Contribution

It constructs nonlinear wave operators and proves asymptotic completeness for the Maxwell-Higgs system on subextremal Kerr spacetimes, including the analysis of scattering maps and their properties.

Findings

Established small-data asymptotic completeness for the system.

Constructed gauge-invariant scattering maps with quadratic expansions.

Proved the framework in the massless case and extended to massive cases under certain conditions.

Abstract

We construct nonlinear wave operators and prove small-data asymptotic completeness for the Maxwell--Higgs system on the domain of outer communications of every four-dimensional subextremal Kerr black hole $(\mathcal D_{M,a},g_{M,a})$ with $M>0$ and $|a|<M$, for gauge-invariant nonnegative scalar potentials $P$ satisfying Assumption~\ref{asumsiP} with mass parameter $m^{2}\ge0$. The massless case $m=0$ is unconditional on the full subextremal range. For $m^{2}>0$ the same conclusions follow assuming the massive linear package $\Lin_{k}^{(m)}$ for the linear comparison system (in particular, no exponentially growing modes); this fails for an open set of masses due to superradiant instability \cite{ShlapentokhRothmanKGKerr}. We work in the radiative (charge-free) regime; stationary Coulomb (Kerr--Newman) modes are treated separately. Asymptotic states are described by gauge-covariant radiation fields on $\mathcal I^{\pm}\cup\mathcal H^{\pm}$ (and, when $m>0$, an additional timelike/Dollard channel), yielding a gauge-invariant nonlinear scattering map on the residual-gauge quotient. The scattering map is a small-data bijection, is Fr\'echet differentiable at $0$ with derivative equal to linear Kerr scattering, admits a quadratic (Born) expansion with an $O(\|U\|^{3})$ remainder in the natural asymptotic topology, and is real-analytic for analytic $P$. The nonlinear argument is presented as a transfer principle from a black-box linear estimate package for inhomogeneous Klein--Gordon and charge-free Maxwell fields, verified here in the massless Kerr case (and proved self-contained in Schwarzschild).