Sur la géométrie de la singularité initiale des espaces-temps plats globalement hyperboliques

Belraouti, Mehdi

Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 457-466.

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 $\ge 2$. 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 $M$ be a maximal globally hyperbolic Cauchy compact flat spacetime of dimension $2+1$, admitting a Cauchy hypersurface diffeomorphic to a compact hyperbolic manifold. We study the asymptotic behaviour of level sets of quasi-concave time functions on $M$. 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.

Classification : 53B30
Mot clés : géométrie lorentzienne, espaces temps à courbure constante, fonction temps quasi-concave, topologie de Gromov équivariante
Keywords: Lorentzian geometry, flat space-time, quasi-concave time function, Gromov equivariant topology

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