On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications

Chru, Piotr T.; Delay, Erwann

Mémoires de la Société Mathématique de France, no. 94 (2003) , 109 p.

Abstract
Résumé

Generalizing an analysis of Corvino and Schoen, we study surjectivity properties of the constraint map in general relativity in a large class of weighted function spaces. As a corollary we prove several perturbation, gluing, and extension results: we show existence of non-trivial, singularity-free, vacuum space-times which are stationary in a neighborhood of $i^{0}$ ; for small perturbations of parity-covariant initial data sufficiently close to those for Minkowski space-time this leads to space-times with a smooth global $ℐ$ ; we prove existence of initial data for many black holes which are exactly Kerr — or exactly Schwarzschild — both near infinity and near each of the connected components of the apparent horizon; under appropriate conditions we obtain existence of vacuum extensions of vacuum initial data across compact boundaries; we show that for generic metrics the deformations in the Isenberg-Mazzeo-Pollack gluings can be localized, so that the initial data on the connected sum manifold coincide with the original ones except for a small neighborhood of the gluing region; we prove existence of asymptotically flat solutions which are static or stationary up to $r^{- m}$ terms, for any fixed $m$ , and with multipole moments freely prescribable within certain ranges.

Nous étudions les propriétés de surjectivité de l’application de contraintes en relativité générale dans une large classe d’espaces fonctionnels à poids, généralisant ainsi une analyse de Corvino et Schoen. Comme corollaire on obtient plusieurs résultats de perturbation, de recollement, ou d’extension. Ainsi, nous démontrons l’existence d’espaces-temps non triviaux, sans singularités, solutions d’équations d’Einstein du vide, qui sont stationnaires dans un voisinage de $i^{0}$ . Pour des données initiales proches de celles de Minkowski ceci conduit, sous une condition de parité approximative, à des espaces-temps avec un infini isotrope $ℐ$ global et lisse. Nous prouvons l’existence de données initiales pour des trous noirs multiples qui sont exactement kerriennes, ou exactement schwarzschildiennes, dans une région asymptotique, mais aussi près de chaque composante connexe de l’horizon apparent. Nous montrons que pour des métriques génériques les perturbations des données initiales introduites par les recollements du type Isenberg-Mazzeo-Pollack peuvent être localisées, de sorte que les données initiales sur la variété obtenue en prenant la somme connexe coincident avec les données initiales originelles, sauf dans un petit voisinage de la zone de recollement. Nous prouvons l’existence de solutions asymptotiquement plates qui sont statiques ou stationnaires modulo des termes en $r^{- m}$ , avec $m$ arbitrairement prescrit, et avec des moments multipolaires qu’on peut spécifier librement dans certains ouverts.

MR: 2031583 | Zbl: 1058.83007 | 3 citations in Numdam

DOI: 10.24033/msmf.407

Classification: 83C05
Keywords: General relativistic initial data, non-connected black holes, asymptotically simple space-times, initial data gluing
Mot clés : Données initiales en relativité générale, trous noirs non connexes, espaces-temps asymptotiquement simples, recollements de données initiales

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Chru, Piotr T.; Delay, Erwann. On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications. Mémoires de la Société Mathématique de France, Serie 2, no. 94 (2003), 109 p. doi : 10.24033/msmf.407. http://numdam.org/item/MSMF_2003_2_94__1_0/

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