{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:55:00Z","timestamp":1777449300437,"version":"3.51.4"},"reference-count":20,"publisher":"SAGE Publications","issue":"1","license":[{"start":{"date-parts":[[2015,12,1]],"date-time":"2015-12-01T00:00:00Z","timestamp":1448928000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2015,12,9]]},"abstract":"<jats:p>We solve the Einstein constraint equations for a [Formula: see text]-dimensional vacuum space\u2013time with a space-like translational Killing field in the asymptotically flat case. The presence of a space-like translational Killing field allows for a reduction of the [Formula: see text]-dimensional problem to a [Formula: see text]-dimensional one. The aim of this paper is to go further in the asymptotic expansion of the solutions than in [Constraint equations for [Formula: see text] vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case, available at: arXiv:1302.1473 ]. In particular the expansion we construct involves quantities which are the 2-dimensional equivalent of the global charges.<\/jats:p>","DOI":"10.3233\/asy-151333","type":"journal-article","created":{"date-parts":[[2015,12,9]],"date-time":"2015-12-09T14:25:19Z","timestamp":1449671119000},"page":"51-89","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":4,"title":["Constraint equations for 3+1 vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case II"],"prefix":"10.1177","volume":"96","author":[{"given":"C\u00e9cile","family":"Huneau","sequence":"first","affiliation":[{"name":"DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK. E-mail:\u00a0"}]}],"member":"179","published-online":{"date-parts":[[2015,12,1]]},"reference":[{"key":"ref001","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.55.669"},{"key":"ref002","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-7953-8_1"},{"key":"ref003","doi-asserted-by":"publisher","DOI":"10.1007\/BF01328358"},{"key":"ref004","doi-asserted-by":"publisher","DOI":"10.1006\/aphy.1995.1012"},{"key":"ref005","doi-asserted-by":"publisher","DOI":"10.1063\/1.524259"},{"key":"ref006","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199230723.001.0001"},{"key":"ref007","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevD.61.084034"},{"key":"ref008","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0011-6_5"},{"key":"ref009","unstructured":"[9]Y.\u00a0Choquet-Bruhat and J.W.\u00a0YorkJr., The Cauchy problem, in: General Relativity and Gravitation, Vol.\u00a01, Plenum, New York, 1980, pp.\u00a099\u2013172."},{"key":"ref010","doi-asserted-by":"publisher","DOI":"10.1515\/9781400863174"},{"key":"ref011","doi-asserted-by":"publisher","DOI":"10.4310\/jdg\/1146169910"},{"key":"ref012","doi-asserted-by":"publisher","DOI":"10.1215\/00127094-1813182"},{"key":"ref013","doi-asserted-by":"publisher","DOI":"10.1007\/s00220-009-0743-2"},{"key":"ref014","unstructured":"[14]C.\u00a0Huneau, Constraint equations for 3+1 vacuum Einstein equations with a translational space-like Killing field in the asymptotically flat case, available at: arXiv:1302.1473."},{"key":"ref015","unstructured":"[15]C.\u00a0Huneau, Stability in exponential time of Minkowski space\u2013time with a translation space-like Killing field, available at: arXiv:1410.6068."},{"key":"ref016","doi-asserted-by":"publisher","DOI":"10.1088\/0264-9381\/12\/9\/013"},{"key":"ref017","first-page":"516","volume":"213","author":"Lichnerowicz A.","year":"1941","journal-title":"C. 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