Sur la g\'eom\'etrie de la singularit\'e initiale des espaces-temps plats globalement hyperboliques

TL;DR

This paper investigates the geometric structure of initial singularities in 2+1 dimensional flat globally hyperbolic spacetimes, demonstrating convergence of certain level sets to a real tree, confirming a conjecture in the field.

Contribution

It proves that level sets of quasi-concave time functions in such spacetimes converge to a real tree, independent of the chosen time function, confirming a conjecture by Benedetti and Guadagnini.

Findings

Level sets converge to a real tree in the Hausdorff-Gromov topology.

Limit is independent of the time function chosen.

Confirms a conjecture by Benedetti and Guadagnini.

Abstract

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 \cite{MR1857817}. 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.