Let MGHC denote the expression “maximal globally hyperbolic spatially compact”. The questions of this paper are those for the existence of \(K\)-time in three-dimensional MGHC space-times of constant curvature, for the existence of Cauchy surfaces of prescribed \(K\)-curvature in three-dimensional MGHC space-times of constant curvature, and for the Minkowski problem in three-dimensional Minkowski space.
Here are some aspects of the contents: relations between the systole of a Cauchy surface \(S\), and the distance from \(S\) to the initial singularity; a proof that the limit of a decreasing sequence of convex Cauchy surfaces is uniformly space-like; constructions of barriers, surfaces with prescribed \(K\)-curvature. An interesting result concerning the existence of \(K\)-slicings of three-dimensional MGHC space-times of constant curvature is proved.
Theorem: Let \(M\) be a three-dimensional non-elementary MGHC space-time with constant curvature \(\Lambda\). If \(\Lambda\geq 0\), reversing the time orientation if necessary, it is assumed that \(M\) is future complete.
\(\bullet\) If \(\Lambda\geq 0\) (flat case or locally de Sitter case), then \(M\) admits a unique \(K\)-slicing. The leaves of this slicing are the level sets of a \(K\)-time ranging over \((-\infty,-\Lambda)\).
\(\bullet\) If \(\Lambda< 0\) (locally anti-de Sitter case), then \(M\) does not admit any global \(K\)-slicing, but each of the two connected components of the complement of the convex core of \(M\) admits a unique \(K\)-slicing. The leaves of the \(K\)-slicing of the past of the convex core are the level sets of a \(K\)-time ranging over \((-\infty,0)\). The leaves of the \(K\)-slicing of the future of the convex core are the level sets of a reverse \(K\)-time ranging over \((-\infty,0)\).
Finally, the authors study the Minkowski problem.
The exposition is clear. The work has a rich bibliography.