Butscher's perturbative method for the construction of initial data sets

The system of Conformal Constraint Equations (CCEs) of H. Friedrich [3] others a promising alternative to the standard Conformal Method for the construction of initial data for the Cauchy problem in GR. As a first step to their analysis, one must first understand the Extended Constraint Equations (ECEs), which can be thought of as a reduction of the CCEs corresponding to a trivial conformal rescaling. While much simpler than the full system, much of the intricate structure of the CCEs is already evident at the level of the ECEs.
In this talk I will outline the pertubative method of A. Butscher [1, 2] for the construction of solutions of the ECEs, with emphasis on its application to the case of closed initial hypersurfaces. Such solutions, constructed as non-linear perturbations of a given \background" initial data set, describe initial data with prescribed extrinsic mean curvature, and prescribed TT parts of the electric and magnetic Weyl curvature. I will give suffcient conditions for the implementation of the method, and some examples of admissible background initial data.
Time permitting, I will describe progress made towards extending the analysis to the full Conformal Constraint Equations.

[1] A. Butscher, Exploring the conformal constraint equations, in The conformal structure of spacetime: Geometry, Analysis, Numerics, edited by J. Frauendiener & H. Friedrich, Lect. Notes. Phys., page 195, 2002.
[2] A. Butscher, Perturbative solutions of the extended constraint equations in General Relativity, Comm. Math. Phys. 272, 1 (2007).
[3] H. Friedrich, Cauchy problems for the conformal vacuum field equations in General Relativity, Comm. Math. Phys. 91, 445 (1983).