Anisotropic Einstein data with isotropic non negative prescribed scalar curvature

Fiedler, Bernold; Hell, Juliette; Smith, Brian

Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 2, pp. 401-428.

Nous construisons des données initiales de trou noir à symétrie temporelle pour les équations d'Einstein dont la courbure scalaire est prescrite ; plus précisément une partie de telles données initiales contenue à l'intérieur du trou noir. Dans ce cas, les contraintes d'Einstein peuvent être exprimées à l'aide d'une équation parabolique dont la variable « temps » est le rayon, vérifiée par une composante u de la métrique qui subit une explosion en « temps » fini. La métrique elle-même est régulière jusqu'à la surface au rayon de l'explosion (inclue) ; cette surface est une surface minimale.Nous montrons l'existence de données vérifiant les contraintes d'Einstein, dont le profil d'explosion est anisotrope (i.e. elles ne sont pas $O\left(3\right)$-symétriques) alors que la courbure scalaire a été prescrite de façon isotrope.Nos résultats sont basés sur la théorie des variétés centrales pour les équations paraboliques quasi-linéaires et sur la théorie équivariante des bifurcations pour des solutions de l'équation en variables auto-similaires, dont l'évolution n'est pas nécessairement auto-similaire.

We construct time-symmetric black hole initial data for the Einstein equations with prescribed scalar curvature, or more precisely a piece of such initial data contained inside the black hole. In this case, the Einstein constraint equations translate into a parabolic equation, with radius as ‘time’ variable, for a metric component u that undergoes blow up. The metric itself is regular up to and including the surface at the blow up radius, which is a minimal surface.We show the existence of Einstein constrained data with blow up profiles that are anisotropic (i.e. not $O\left(3\right)$ symmetric) although the scalar curvature was isotropically prescribed.Our results are based on center manifold theory for quasilinear parabolic equations and on equivariant bifurcation theory for not necessarily self-similar solutions of a self-similarly rescaled equation.

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