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Annals of PDE

, 4:2 | Cite as

An Extension Procedure for the Constraint Equations

  • Stefan Czimek
Manuscript
  • 67 Downloads

Abstract

Let \((\bar{g}, \bar{k})\) be a solution to the maximal constraint equations of general relativity on the unit ball \(B_1\) of \({\mathbb R}^3\). We prove that if \((\bar{g},\bar{k})\) is sufficiently close to the initial data for Minkowski space, then there exists an asymptotically flat solution (gk) on \({\mathbb R}^3\) that extends \((\bar{g}, \bar{k})\). Moreover, (gk) is bounded by \((\bar{g}, \bar{k})\) and has the same regularity. Our proof uses a new method of solving the prescribed divergence equation for a tracefree symmetric 2-tensor, and a geometric variant of the conformal method to solve the prescribed scalar curvature equation for a metric. Both methods are based on the implicit function theorem and an expansion of tensors based on spherical harmonics. They are combined to define an iterative scheme that is shown to converge to a global solution (gk) of the maximal constraint equations which extends \((\bar{g},\bar{k})\).

Keywords

General relativity Constraint equations Analysis of PDE Differential geometry 

Notes

Acknowledgements

This work forms part of my Ph.D. thesis and I am grateful to my Ph.D. advisor Jérémie Szeftel for his kind supervision and careful guidance. Furthermore, I am grateful to the RDM-IdF for financial support.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsUniversité Pierre et Marie Curie (Paris 6)ParisFrance

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