Asymptotic behavior of Cauchy hypersurfaces in constant curvature space-times

TL;DR

This paper investigates the asymptotic behavior of convex Cauchy hypersurfaces in various constant curvature space-times, showing convergence to real trees or CAT(0) spaces as time approaches zero, independent of the chosen time function.

Contribution

It generalizes previous results to (2+1) de Sitter and anti de Sitter cases and extends the analysis to flat (n+1) dimensions, establishing convergence to specific geometric limits.

Findings

Level sets converge to a real tree in de Sitter and anti de Sitter cases.

Level sets converge to a CAT(0) space in flat case.

Limit behavior is independent of the choice of time function.

Abstract

We study the asymptotic behavior of convex Cauchy hypersurfaces on maximal globally hyperbolic spatially compact space-times of constant curvature. We generalise the result of [11] to the (2+1) de Sitter and anti de Sitter cases. We prove that in these cases the level sets of quasi-concave times converge in the Gromov equivariant topology, when time goes to 0, to a real tree. Moreover, this limit does not depend on the choice of the time function. We also consider the problem of asymptotic behavior in the flat (n+1) dimensional case. We prove that the level sets of quasi-concave times converge in the Gromov equivariant topology, when time goes to 0, to a CAT (0) metric space. Moreover, this limit does not depend on the choice of the time function.