Solutions of the Einstein equations with cylindrical symmetry
which are asymptotically flat in all directions allowed by the
symmetry represent an interesting variation on asymptotic
flatness. Since black holes are apparently incompatible with this
symmetry, one may hope to prove geodesic completeness of
solutions under appropriate assumptions. (It would be interesting
to have a theorem making the statement about black holes
precise.) A proof of geodesic completeness has been achieved for
the Einstein vacuum equations and for the source-free
Einstein-Maxwell equations in [22], building on global existence theorems for wave maps [57,
56]. For a quite different point of view on this question involving
integrable systems see [167]. A recent preprint of Hauser and Ernst [91] also appears to be related to this question. However, due to
the great length of this text and its reliance on many concepts
unfamiliar to this author, no further useful comments on the
subject can be made here.