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

TL;DR

This paper extends a perturbative method for constructing solutions to Einstein's vacuum constraint equations on compact manifolds, allowing for controlled nonlinear perturbations of specific initial data.

Contribution

It adapts Friedrich-Butscher's perturbative approach to compact manifolds, providing explicit parametrization of free data and proving local existence of solutions.

Findings

Existence of solutions near constant mean curvature data with negative curvature.

Explicit parametrization of free data components.

Construction of nonlinear perturbations of initial data.

Abstract

We discuss the implementation, to the case of compact manifolds, of the perturbative method of Friedrich-Butscher for the construction of solutions to the vaccum Einstein constraint equations. This method is of a perturbative nature and exploits the properties of the extended constraint equations ---a larger system of equations whose solutions imply a solution to the Einstein constraints. The method is 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.