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

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- Written by Jarrod Williams

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.

References

[1] A. Butscher,* Exploring the conformal constraint equation*s, 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).