In the 90’s, in a celebrated preprint, G. Mess described all
lorentzian metric with constant curvature on the manifold
S x [-1, 1] where S is a closed surface.
From then, it has been understood how to generalize part of this
description
to higher dimensions (but still in constant curvature),
in terms of a classic notion in general relativity: the notion
of ital{maximal globally hyperbolic} (abbrev. MGH) spacetimes.
Roughly speaking, a MGH spacetime is a (time oriented)
lorentz manifold (M,g) admitting a function t: M —> R
which is increasing along any future oriented causal curve, and such that
the restriction of t to any ital{inextendible} causal curve is surjective.
I will try to motivate this definition of MGH spacetimes; present what
is known about the classification of MGH spacetimes of constant curvature,
and present my recent joint work with L. Andersson, F. Béguin, A. Zeghib
where we show that (non-elementary) spatially compact MGH spacetimes admits
a
time function t such that every fiber of t has constant mean curvature.