Annals of PDE

, 4:2 | Cite as

An Extension Procedure for the Constraint Equations

  • Stefan CzimekEmail author


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 \((\bar{g},\bar{k})\).


General relativity Constraint equations Analysis of PDE Differential geometry 



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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Authors and Affiliations

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

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