Anisotropic Einstein data with isotropic non negative prescribed scalar curvature
Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 2, p. 401-428
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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(3) 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.

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(3)-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.

@article{AIHPC_2015__32_2_401_0,
     author = {Fiedler, Bernold and Hell, Juliette and Smith, Brian},
     title = {Anisotropic Einstein data with isotropic non negative prescribed scalar curvature},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {32},
     number = {2},
     year = {2015},
     pages = {401-428},
     doi = {10.1016/j.anihpc.2014.01.002},
     zbl = {1332.35358},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2015__32_2_401_0}
}
Fiedler, Bernold; Hell, Juliette; Smith, Brian. Anisotropic Einstein data with isotropic non negative prescribed scalar curvature. Annales de l'I.H.P. Analyse non linéaire, Volume 32 (2015) no. 2, pp. 401-428. doi : 10.1016/j.anihpc.2014.01.002. http://www.numdam.org/item/AIHPC_2015__32_2_401_0/

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