The Constraint Equations in the Presence of a Scalar Field: The Case of the Conformal Method with Volumetric Drift

In this paper we establish the existence in low dimensions of solutions to the constraint equations in the case of the conformal system recently proposed by Maxwell (Initial data in general relativity described by expansion, conformal deformation and drift, 2014. arXiv:1407.1467v1), with the added presence of a scalar field and under suitable smallness assumptions on its parameters.