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 ; 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 ; 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 terms, for any fixed , 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 . 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 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 , avec arbitrairement prescrit, et avec des moments multipolaires qu’on peut spécifier librement dans certains ouverts.
Keywords: General relativistic initial data, non-connected black holes, asymptotically simple space-times, initial data gluing
@book{MSMF_2003_2_94__1_0,
author = {Chru, Piotr T. and Delay, Erwann},
title = {On mapping properties of the~general relativistic constraints operator in~weighted function spaces, with~applications},
series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
publisher = {Soci\'et\'e math\'ematique de France},
number = {94},
year = {2003},
doi = {10.24033/msmf.407},
zbl = {1058.83007},
mrnumber = {2031583},
language = {en},
url = {http://archive.numdam.org/item/MSMF_2003_2_94__1_0/}
}
TY - BOOK AU - Chru, Piotr T. AU - Delay, Erwann TI - On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications T3 - Mémoires de la Société Mathématique de France PY - 2003 DA - 2003/// IS - 94 PB - Société mathématique de France UR - http://archive.numdam.org/item/MSMF_2003_2_94__1_0/ UR - https://zbmath.org/?q=an%3A1058.83007 UR - https://www.ams.org/mathscinet-getitem?mr=2031583 UR - https://doi.org/10.24033/msmf.407 DO - 10.24033/msmf.407 LA - en ID - MSMF_2003_2_94__1_0 ER -
%0 Book %A Chru, Piotr T. %A Delay, Erwann %T On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications %S Mémoires de la Société Mathématique de France %D 2003 %N 94 %I Société mathématique de France %U https://doi.org/10.24033/msmf.407 %R 10.24033/msmf.407 %G en %F MSMF_2003_2_94__1_0
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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