On étudie le comportement asymptotique des niveaux d’une fonction temps quasi-concave, définie sur un espace-temps globalement hyperbolique maximal plat de dimension trois, admettant une hypersurface de Cauchy de genre . On donne une réponse positive à une conjecture posée par Benedetti et Guadagnini dans [7]. Plus précisément, on montre que les niveaux d’une telle fonction temps convergent au sens de la topologie de Hausdorff-Gromov équivariante vers un arbre réel. On montre de plus que la limite est indépendante de la fonction temps choisie.
Let be a maximal globally hyperbolic Cauchy compact flat spacetime of dimension , admitting a Cauchy hypersurface diffeomorphic to a compact hyperbolic manifold. We study the asymptotic behaviour of level sets of quasi-concave time functions on . We give a positive answer to a conjecture of Benedetti and Guadagnini in [7]. More precisely, we prove that the level sets of such a time function converge in the Hausdorff-Gromov equivariant topology to a real tree. Moreover, this limit does not depend on the choice of the time function.
@article{AIF_2014__64_2_457_0,
author = {Belraouti, Mehdi},
title = {Sur la g\'eom\'etrie de la singularit\'e initiale des espaces-temps plats globalement hyperboliques},
journal = {Annales de l'Institut Fourier},
volume = {64},
year = {2014},
pages = {457-466},
doi = {10.5802/aif.2854},
zbl = {06387281},
mrnumber = {3330911},
language = {fr},
url = {http://dml.mathdoc.fr/item/AIF_2014__64_2_457_0}
}
Belraouti, Mehdi. Sur la géométrie de la singularité initiale des espaces-temps plats globalement hyperboliques. Annales de l'Institut Fourier, Tome 64 (2014) pp. 457-466. doi : 10.5802/aif.2854. http://gdmltest.u-ga.fr/item/AIF_2014__64_2_457_0/
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