Solution of constraints for Einstein equations on non maximal asymptotically Euclidean manifolds. (Solution des contraintes pour les équations d’Einstein sur une variété asymptotiquement euclidienne non maximale.) (French) Zbl 0783.53047
Summary: We consider the coupled elliptic system of constraints on the Cauchy data for Einstein’s equations obtained by the conformal method when the initial manifold is asymptotically Euclidean and has a non-zero mean extrinsic curvature \(\tau\). We prove that this system admits asymptotically Euclidean solutions when \(\nabla\tau\) is small in an appropriate norm and the scalar curvature of the conformal metric \(g\) is either everywhere negative with fall off conditions that we specify, or such that \(g\) belongs to the positive Yamabe class.
MSC:
53Z05 | Applications of differential geometry to physics |
83C20 | Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory |