On smoothness of timelike maximal cylinders in three dimensional vacuum   spacetimes
  We show that timelike maximal cylinders in $\RR^{1 + 2}$ always develop singularities in finite time and that, infinitesimally at a generic singularity, their time slices are evolved by a rigid motion or a self-similar motion. We also prove a mild generalization in non-flat backgrounds. 
Luc Nguyen and Gang Tian
math.DG gr-qc