A perturbative approach to the construction of initial data on compact manifolds

We discuss the implementation of Butscher's method \cite{But06,But07} for the construction of solutions to the extended constraint equations, for the vacuum Einstein field equations, in the case of compact manifolds. The method is then applied to the construction of nonlinear perturbations of constant mean curvature initial data of constant negative sectional curvature. We prove the existence of a neighbourhood of solutions to the constraint equations around such initial data, with particular components of the extrinsic curvature and electric/magnetic parts of the spacetime Weyl curvature prescribed as free data. The space of such free data is parametrised explicitly.