Scattering for the Equivariant U(1) Problem

Extending our previous works on the Cauchy problem for the \(2+1\) $2+1$ equivariant Einstein-wave map system, we prove that the linear part dominates the nonlinear part of the wave maps equation coupled to the full set of the Einstein equations, for small data. A key ingredient in the proof is a nonlinear Morawetz estimate for the fully coupled equivariant Einstein-wave maps. The \(2+1\) $2+1$ -dimensional Einstein-wave map system occurs naturally in the \(3+1\) $3+1$ vacuum Einstein equations of general relativity.